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TRANSLATION SURFACES AND THE CURVE GRAPH IN GENUS TWO

DUC-MANH NGUYEN

Abstract. Let S be a (topological) compact closed surface of genus two. We associate to each transla-tion surface (X, ω) ∈ ΩM2 = H(2)⊔ H(1,1) a subgraphCcyl of the curve graph ofS. The vertices ofthis subgraph are free homotopy classes of curves which can be represented either by a simple closedgeodesic, or by a concatenation of two parallel saddle connections (satisfying some additional proper-ties) onX. The subgraphCcyl is by definition GL+(2,R)-invariant. Hence it may be seen as the imageof the corresponding Teichmüller disk in the curve graph. Wewill show thatCcyl is always connectedand has infinite diameter. The group Aff+(X, ω) of affine automorphisms of (X, ω) preserves naturallyCcyl, we show that Aff+(X, ω) is precisely the stabilizer ofCcyl in Mod(S). We also prove thatCcyl isGromov-hyperbolic if (X, ω) is completely periodic in the sense of Calta.

It turns out that the quotient ofCcyl by Aff+(X, ω) is closely related to McMullen’s prototypes inthe case (X, ω) is a Veech surface inH(2). We finally show that this quotient graph has finitely manyvertices if and only if (X, ω) is a Veech surface for (X, ω) in both strataH(2) andH(1, 1).

1. Introduction

1.1. Curve complex. Let S be compact surface. Thecurve complexof S is a simplicial complexwhose vertices are free homotopy classes of essential simple closed curves onS, andk-simplices aredefined to be the sets of (homotopy classes of)k+ 1 curves that can be realized pairwise disjointly onS. This complex was introduced by Harvey [22] in order to use its combinatorial structure to encodethe asymptotic geometry of the Teichmüller space. It turns out that its geometry is intimately related tothe geometry and topology of Teichmüller space (seee.g. [50]). The curve complex has now becomea central subject in Teichmüller Theory, Low Dimensional Topology, and Geometric Group Theory.Note that this complex is quasi-isometric to its 1-skeletonwhich is referred to as thecurve graphofS. In this paper we will denote the curve graph byC(S).

The Mapping Class Group Mod(S) naturally acts on the curve complex by isomorphisms. Inmost cases, all automorphisms of the curve complex are induced by elements of Mod(S) (see [26,33]). Based on this relation, topological and combinatorial properties of the curve complex havebeen used to study the mapping class group ([21, 2]). In [35],Masur and Minsky showed that thecurve graph (and the curve complex) is Gromov-hyperbolic (see also [5]). A stronger result, that isthe hyperbolicity constant is independent of the surfaceS, has been proved recently simultaneouslyby several people [1, 7, 12, 23]. Its boundary at infinity has been studied by Klarreich [29] andHamenstädt [18]. Those results have led to numerous applications and a fast growing literature on thesubject. In particular, the hyperbolicity of the curve graph has been exploited in the resolution of theEnding Lamination Conjecture by Brock-Canary- Minsky [8].For a nice survey on the curve complexand its applications we refer to [6].

Date: June 25, 2018.1

2 DUC-MANH NGUYEN

1.2. Teichmüller disk and translation surface. Another important notion in Teichmüller theory isthe Teichmüller disks. These are isometric embeddings of the hyperbolic diskH in the Teichmüllerspace. Such a disk can be viewed as a complex geodesic generated by a quadratic differentialq ona Riemann surfaceX. This quadratic differential defines a flat metric structure onX with conicalsingularities such that the holonomy of any closed curve onX belongs to the subgroup±Id × R2

of Isom(R2). If this quadratic differential is the square of a holomorphic one formω on X, then theholonomy of any closed curve is a translation ofR2, and we have a translation surface (X, ω).

Using the flat metric viewpoint, one can easily define a natural action of GL+(2,R) on the space oftranslation surfaces as follows: given a matrixA ∈ GL+(2,R) and an atlasφi , i ∈ I defining a transla-tion surface structure, we get an atlas for another translation surface structure defined byAφi , i ∈ I .The Teichmüller disk generated by a holomorphic one-form (X, ω) is precisely the projection into theTeichmüller space of its GL+(2,R)-orbit. Translations surfaces and their GL+(2,R)-orbit also arise indifferent contexts such as dynamics of billiards in rational polygons, interval exchange transforma-tions, pseudo-Anosov homeomorphisms...

The importance of the GL+(2,R)-action is related to the fact that the GL+(2,R)-orbit closure ofa translation surface encodes information on its geometricand dynamical properties. A remarkableillustration of this phenomenon is the famous Veech’s Dichotomy, which states that if the stabilizer of(X, ω) for the action of GL+(2,R) is a lattice in SL(2,R), then the linear flow in any direction onX iseither periodic or uniquely ergodic. Following a work of Smillie, the stabilizer of (X, ω), denoted bySL(X, ω), is a lattice in SL(2,R) if and only if the GL+(2,R)-orbit of (X, ω) is a closed subset of themoduli space. For more details on translation surfaces and related problems we refer to the excellentsurveys [37, 57].

The group SL(X, ω) is closely related to the subgroup of the mapping class group that stabilizes theTeichmüller disk generated by (X, ω). This subgroup consists of elements of Mod(S) that are realizedby homeomorphisms ofX preserving the set of singularities (for the flat metric), and given by affinemaps in local charts of the flat metric structure. This subgroup is denoted by Aff+(X, ω). There is anatural homomorphism from Aff+(X, ω) to SL(X, ω) which associates to each element of Aff+(X, ω)its derivative. It is not difficult to see that this homomorphism is surjective and has finite kernel. Thestudy of Aff+(X, ω) and SL(X, ω) is a recurrent theme in the theory of dynamics in Teichmüller space(seee.g. [38, 24, 25, 44, 32]).

1.3. Flat metric and curve complex. Consider now the flat metric defined by a holormorphic one-formω on a (compact) Riemann surfaceX. By compactness, there exists a curve of minimal length inthe free homotopy class of any essential simple closed curve. In general this curve of minimal lengthmay not be a geodesic as it may contain some singularity in itsinterior. Nevertheless, following aresult by Masur [34], we know that there are infinitely many curves that can be realized as simpleclosed geodesics forω. Thus (X, ω) specifies a subset of vertices ofC(S). Note that unlike thesituation of hyperbolic surface, closed geodesics of minimal length are not unique in their homotopyclass. They actually arise in family, that is simple closed geodesics in the same homotopy class fill outa subset ofX which is isometric to (R/cZ) × (0, h). We will call such a subset ageometric cylinder,and the corresponding simple closed geodesics itscore curves.

Mimicking the construction of the curve graph, we can add an edge between two vertices rep-resenting two cylinders if there exist two curves, one in each homotopy class, that can be realized

TRANSLATION SURFACES AND THE CURVE GRAPH IN GENUS TWO 3

disjointly (this condition is equivalent to requiring thatthe corresponding geodesics for the flat metricare disjoint). Thus, for each translation surface, we have asubgraphCcyl of the curve graph.

Let A be a matrix in GL+(2,R), and consider the surface (X′, ω′) := A · (X, ω). Since the actionof A preserves the affine structure, a geodesic onX corresponds to a geodesic onX′ and vice-versa.Therefore, the subgraphs associated to (X′, ω′) and to (X, ω) are the same. This subgraph is actuallyassociated to the Teichmüller disk generated by (X, ω). As C(S) can be viewed as the combinatorialmodel for the Teichmüller space,Ccyl can be viewed as the counterpart of a Teichmüller disk in thissetting. By definition, elements of Aff+(X, ω) preserveCcyl and act onCcyl by isomorphisms. Asproperties of the Mapping Class Group can be studied via its action on the curve complex, one canexpect the knowledge about the combinatorial and geometricstructure ofCcyl to be useful for thestudy of Aff+(X, ω).

1.4. Statement of results. The main purpose of this paper is to investigateCcyl whenX is a surface ofgenus two. The reason for this restriction is the technical difficulties for the general cases. Hopefully,the results and techniques used in this situation may inspire further results in higher genera.

Recall that the moduli space of translation surfaces is naturally stratified by the zero orders of theone-formω (or equivalently, the cone angles at the singularities). Ingenus two, we have two strata:H(2) which contains pairs (X, ω) such thatω has a unique double zero, andH(1, 1) which containspairs (X, ω) such thatω has two simple zeros. Our first result shows that the geometryof Ccyl doesdepend on the stratum of (X, ω).

Theorem A (Theorem 2.6). If (X, ω) ∈ H(2) thenCcyl contains no triangles, but if(X, ω) ∈ H(1, 1)thenCcyl always contains some triangles.

Note that a triangle inCcyl is a triple of simple closed curves pairwise disjoint that are simultane-ously realized as core curves of three cylinders in (X, ω).

From its definition, the geometric structure of the subgraphCcyl depends very much on the flatmetric of (X, ω). It is not difficult to see thatCcyl is not connected in general (see Section 3). To get a“nicer” subgraph ofC(S), we enlargeCcyl by adjoin to it the vertices ofC(S) representingdegeneratecylinders. Roughly speaking, a degenerate cylinder onX is a union of two saddle connections in thesame direction such that there are deformations of (X, ω), on which this union is freely homotopicto the core curves of a geometric cylinder. We refer to Section 3 for a more precise definition. Inparticular, any degenerate cylinder is freely homotopic a simple closed curve. Thus it corresponds toa vertex ofC(S).

We defineC(0)cyl to be the set of vertices ofC(S) that correspond to geometric cylinders and degen-

erate cylinders in (X, ω). We then defineC(1)cyl to be the set of the edges ofC(S) whose both endpoints

belong toC(0)cyl. We thus get a subgraphCcyl of C(S). By a slight abuse of notation, we will also call

Ccyl thecylinder graphof (X, ω). Subsequently, this subgraph is the main object of our investigation.We resume the results concerningCcyl in the following

Theorem B. For any (X, ω) ∈ H(1, 1) ⊔ H(2), the subgraphCcyl is connected and has infinitediameter. The subgroup ofMod(S) that stabilizesCcyl is preciselyAff+(X, ω). Moreover, if(X, ω) iscompletely periodic in the sense of Calta thenCcyl is Gromov-hyperbolic.

4 DUC-MANH NGUYEN

Theorem B actually comprises several statements which are proved in Corollary 4.2, Proposi-tion 5.1, Proposition 6.1, and Theorem 7.1. The contexts andprecise statements will be given inthe corresponding sections.

We finally consider the quotient ofCcyl by the action of Aff+(X, ω) in the case (X, ω) is a Veechsurface (that is SL(X, ω) is a lattice of SL(2,R)).

Theorem C. Let G be the quotient ofCcyl by the group of affine automorphisms. Then(X, ω) ∈H(2)⊔ H(1, 1) is a Veech surface if and only ifG has finitely many vertices. For any Veech surfacein H(2) the set of edges ofG is also finite. There exist Veech surfaces inH(1, 1) such thatG hasinfinitely many edges.

The statements of Theorem C are proved in Theorem 8.1 and Proposition 8.2.

1.5. Outline. The paper is organized as follows:In Section 2 we recall standard notions concerning translation surfaces. We show some geometric

and topological features of translation surfaces of genus two. We end this section by the proof ofTheorem A.

In Section 3, we introduce the notion of degenerate cylinders and define the cylinder graphsCcyl

andCcyl. We show thatCcyl is connected and has infinite diameter in Section 4 and Section 5. Thoseresults follow from Theorem 4.1 which gives a bound on the distance inCcyl using the intersectionnumber.

Section 6 is devoted to the proof of the fact that the stabilizer subgroup ofCcyl in Mod(S) isprecisely the group of affine automorphisms.

In Section 7 we show that if (X, ω) is completely periodic in the sense of Calta thenCcyl is Gromov-hyperbolic. Our proof follows a strategy of Bowditch and uses a hyperbolicity criterion by Masur-Schleimer.

We give the proof of Theorem C in Section 8. Finally, in Section 9, we give the connection betweenthe quotient graphG = Ccyl/Aff

+ and the set of prototypes for Veech surfaces inH(2), which wereintroduced by McMullen [39].

1.6. Acknowledgements.The author warmly thanks Arnaud Hilion for the very helpful and stimu-lating discussions.

2. Preliminaries

In this section we will prove some topological properties ofsaddle connections and cylinders ontranslation surfaces in genus two. The main result of this section is Theorem 2.6.

Let (X, ω) be a translation surface. A saddle connection onX is a geodesic segments whose end-points are singularities which contains no singularities in its interior. A (geometric) cylinder ofX isa subsetC isometric to (R/cZ) × (0, h), with c, h ∈ R>0, which is not properly contained in anothersubset with the same property. The parameterc is called thecircumferenceandh thewidth or heightof this cylinder.

The isometry from (R/cZ)× (0, h) to C can be extended by continuity to a map from (R/cZ)× [0, h]to X. We will call the images of (R/cZ) × 0 and (R/cZ) × h the boundary components ofC.Each boundary component is a concatenation of some saddle connections. It may happen that the

TRANSLATION SURFACES AND THE CURVE GRAPH IN GENUS TWO 5

two boundary components coincide as subsets ofX. We say thatC is simple cylinderif each of itsboundary component is a single saddle connection. It is worth noticing that on a translation surfaceof genus two, every cylinder is invariant by the hyperelliptic involution. Therefore the two boundarycomponents of any cylinder contain the same number of saddleconnections.

Throughout this paper, for any cyclec ∈ H1(X, zeros ofω;Z), we will use the notationω(c) :=∫cω, and for any saddle connections, its euclidean length will be denoted by|s|. Let us start by the

following elementary lemma.

Lemma 2.1. Let (X, ω) be a translation surface in one of the hyperelliptic componentsHhyp(2g− 2)orHhyp(g−1, g−1), and s be a saddle connection invariant by the hyperellipticinvolutionτ of X. Weassume that s is not vertical. Then there exist a parallelogram P = (P1P2P3P4) in R2, and a locallyisometric mappingϕ : P→ X such that

a) The vertical lines through the vertices P3 and P4 intersect the diagonalP1P2.b) The vertices ofP are mapped to the singularities of X, andP1P2 is mapped isometrically to s.c) The restriction ofϕ into int(P) is an embedding.d) Letη > 0 be the length of the vertical segment from P3 or P4 to a point inP1P2. Then for any

vertical segment u in X from a singular point to a point in s, wehave|u| ≥ η, where|u| is theeuclidian length of u.

We will call P theembedded parallelogramassociated to s.

Remark 2.2.

• Since s in invariant byτ, we must haveτ(ϕ(P)) = ϕ(P).• The sides ofP are mapped to saddle connections on X. Even though the restriction ofϕ into

int(P) is one-to-one, it may happen thatϕ maps the opposite sides ofP to the same saddleconnection.• This lemma is also valid for translation surfaces inH(0) andH(0, 0).

Proof of Lemma 2.1.We will only give the proof for the case (X, ω) ∈ Hhyp(2g − 2) as the proof forHhyp(g− 1, g− 1) is the same. UsingU− =

(1 0t 1

), t ∈ R, we can assume thats is horizontal. LetΨt

be the vertical flow onX generated by the vertical vector field (0, 1), this flow moves regular points ofX vertically, upward ift > 0.

Consider the vertical geodesic rays emanating from the unique zeroP0 of ω in direction (0,−1).We claim that one of the rays in this direction must meets. Indeed, if this is not the case then for anyt ∈ R>0, Ψt(s) does not containP0, and it follows that one can embed a rectangle of infinite areaintoX. Let u+ be a vertical geodesic segment of minimal length fromP0 to a point inswhich is includedin a ray in direction (0,−1). Sinces is invariant byτ, u− := τ(u+) is a vertical segment of minimallength fromP0 to a point ins which is included in a ray in direction (0, 1). Using the developingmap, we can realizes as a horizontal segmentP1P2 ⊂ R2, u+ (resp.u−) as a vertical segmentP3P′3(resp.P4P′4) whereP′3,P′4 ∈ P1P2. Remark that the central symmetry fixing the midpoint ofP1P2

exchangesP3P′3 andP4P′4.Let P denote the parallelogram (P1P3P2P4). We define a mapϕ : P → X as follows: for any

point M ∈ P, let M′ be the orthogonal projection ofM in P1P2, and t be the length ofMM′. LetM′ be the point ins corresponding toM′ by the identification betweenP1P2 and s. We then define

6 DUC-MANH NGUYEN

ϕ(M) := Ψt(M′) if M is aboveP1P2, andϕ(M) = Ψ−t(M′) if M is belowP1P2. By definition,ϕ is alocal isometry and maps the vertices ofP to P0.

Note that we have|MM′| ≤ |P3P′3| = |P4P′4| and the equality only occurs whenM = P3, orM = P4. Thus, for allM ∈ P \ P1,P2,P3,P4, ϕ(M) is a regular point inX (otherwise we wouldhave a vertical segment fromP0 to a point insof length smaller than|u+|).

We now claim thatϕ|int(P) is an embedding. Assume that there exist two pointsM1,M2 ∈ int(P)

such thatϕ(M1) = ϕ(M2). Set−→v :=−−−−−→M1M2, then for anyM,M′ ∈ P such that

−−−−→MM′ = −→v , we

haveϕ(M) = ϕ(M′). SinceP is a parallelogram, there exists a vertexPi ∈ P1,P2,P3,P4 and a

point M′ ∈ P \ P1,P2,P3,P4 such that−−−−→Pi M′ =

−→v , which implies thatϕ(M′) = P0, and we have acontradiction to the observation above.

It is now straightforward to verify thatP andϕ satisfy all the required properties.

In what follows, by aslit torus we will mean a triple (X, ω, s) whereX is an elliptic curve,ω anon-zero holomorphic one-form, andsan embedded geodesic segment (with respect to the flat metricdefined byω) on X. The following lemma is useful for us in the sequel.

Lemma 2.3. Let (X, ω, s) be a slit torus and P1,P2 be the endpoints of s. Assume that the segment(slit) s is not vertical, that is|Reω(s)| > 0. Then there exist a pair of parallel simple closed geodesicsc1, c2 with ci passing through Pi such that ci ∩ int(s) = ∅, and 0 ≤ |Reω(ci)| ≤ |s|. In particular,the geodesics c1, c2 cut X into two cylinders, one of which containsint(s). Moreover, any leaf of thevertical foliation intersecting ci must intersect s, and if every leaf of the vertical foliationmeets s, thenwe have|Reω(ci)| > 0.

Proof. Remark that a flat torus with two marked points can be considered as hyperelliptic transla-tion surface. Here, the hyperelliptic involution is the unique one that acts by−Id on H1(X,Z) andexchanges the marked points. Therefore, this lemma is a particular case of Lemma 2.1.

We now turn into translation surfaces in genus two. Let (X, ω) be a translation surface inH(2) ⊔H(1, 1). We denote byτ the hyperelliptic involution ofX.

Lemma 2.4. Let s1, s2 be a pair of saddle connections in X which are permuted byτ. If (X, ω) ∈ H(2),then s1 and s2 bound a simple cylinder. If(X, ω) ∈ H(1, 1) then we have two cases:

• if si joins a zero ofω to itself then s1 and s2 bound a simple cylinder,• if si joins two different zeros ofω then s1 ∪ s2 decomposes X as a connected sum of two slit

tori.

Proof. Sinceτ acts by−Id on H1(X,Z), s1 ands2 must be homologous. This lemma follows from aninspection on the configurations of rays originating from the zero(s) ofω in the same direction.

Lemma 2.5. Let (X, ω) be a surface inH(2) and s be a saddle connection in X invariant by thehyperelliptic involutionτ. Then there exist two disjoint cylinders C1,C2 that do not intersect s (thatis, C1 ∩ C2 = ∅, and the core curves of C1 and C2 do not meet s). Remark that s may be containedin the boundary of C1 or C2. The possible configurations of C1 and C2 with respect to s are shown inFigure 1.

Proof. Without loss of generality, we can assume thats is horizontal. LetP = (P1P3P2P4) be theembedded parallelogram associated tos, andϕ : P→ X be the embedding map such thats= ϕ(P1P2)

TRANSLATION SURFACES AND THE CURVE GRAPH IN GENUS TWO 7

s

C1 C2

(a)

C1

C2

s

s

(b)

Figure 1. Configurations ofC1,C2 with respect tos: (a) none ofC1,C2 containss inits boundary, (b)s is contained in the boundary ofC2.

(see Lemma 2.1). We choose the labeling of the vertices ofP such thatP3 is the highest vertex, andP4 is the lowest one. Throughout the proof, we will refer to Figure 2.

Let d+1 = ϕ(P3P1), d+2 = ϕ(P3P2), d−1 = ϕ(P4P2), d−2 = ϕ(P4P1). We haved−i = τ(d+

i ). ByLemma 2.4, eitherd+i = d−i as subsets ofX, or d±i bound a simple cylinder. Remark thatd+1 andd+2cannot be both invariant byτ, otherwise we would haveX = ϕ(P), andX must be a torus. Thus weonly have to consider two cases:

i) Both d±1 andd±2 are respectively boundaries of two simple cylindersC1,C2 in X. In this case,it is not difficult to see that bothC1 andC2 are disjoint fromϕ(P), andC1 ∩C2 = ∅. We thenget the configuration (a).

ii) One of d+1 , d+

2 is invariant byτ, the other bounds a simple cylinder. In this case,ϕ(P) isactually a simple cylinder. Without loss of generality, we can assume thatd±1 bound thecylinderC = ϕ(P), andτ(d+2 ) = d−2 .

Let P5 be the point inR2 such that the triangle (P3P5P2) is the image of (P1P2P4) by the

translation by−−−−→P1P3. Using the assumption thatϕ(P3P2) = ϕ(P1P4), we see thatϕ extends

to a local isometric map fromP′ = (P1P2P5P3) to X such thatϕ(P′) = C andϕ|int(P′) is anembedding (see Figure 2).

Consider the horizontal rays emanating from the unique zerox0 of ω to the outside ofC.By the same argument as in Lemma 2.1, we see that one of the raysin direction (1, 0) reachesd+1 = ϕ(P3P1) from the outside ofC. It follows that we can then extendϕ to a convex hexagonH := (P1P2Q2P5P3Q1), which is the union ofP′ and two triangles (P2Q2P5) and (P3Q1P1).Note that (P2Q2P5) and (P3Q1P1) are exchanged by the central symmetry fixing the midpointof P2P3, and all the vertices ofH are mapped tox0.

Let d+3 = ϕ(P3Q1), d+4 = ϕ(Q1P1), d−3 = ϕ(P2Q2), d−4 = ϕ(Q2P5). Again, for i = 3, 4,we have eitherd+i = d−i , or d±i bound a simple cylinder. Ifd+i = d−i for both i = 3, 4, thenX = ϕ(H) andX must be a flat torus, so we have a contradiction. If bothd±i are the boundariesof simple cylinders, then these cylinders are disjoint, andalso disjoint fromϕ(H). It followsthat the total angle atx0 is at least 8π (the total angle ofH plus 4π), thus we have again acontradiction. We can then conclude that one of the pairsd±3 , d

±4 consist of a single saddle

connection, and the other pair bound a simple cylinder. Without loss of generality, we canassume thatd±3 bound a simple cylinderC3, andd+4 = d−4 = d4. Note thatC3 must be disjointfrom ϕ(H), and in particular it is disjoint froms.

8 DUC-MANH NGUYEN

P1 P2

Q2

P5P3

Q1

P4

s

s

d+4

d−4

d−3

d+3

d−2d−1

D

Figure 2. Finding a cylinder disjoint froms.

Let d+ = ϕ(Q1P5), d− = ϕ(P1Q2) then d± is the boundary of a cylinderD whose corecurves crossd±4 . If H is strictly convex thenD is a simple cylinder, but ifP2Q2 is parallelto P1P2 then D is not simple (in this case we actually haveD = ϕ(H)). Nevertheless, inboth cases the core curves ofD do not intersects. SinceD is contained inϕ(H), we haveC3 ∩ D = ∅. Since bothC3 andD are disjoint froms, the lemma is proved.

We are now ready to show

Theorem 2.6.a) On any(X, ω) ∈ H(2) there always exist two disjoint simple cylinders. There cannot exist a

triple of pairwise disjoint cylinders in X.b) On any(X, ω) ∈ H(1, 1) there always exists a triple of cylinders which are pairwise-disjoint.

Remark 2.7.• The cylinders in Theorem 2.6 are not necessarily parallel.• There cannot exist more than 3 simple closed curves pairwisedisjoint on S . The statement b)

means that given any holomorphic one-form inH(1, 1), there always exists a family of threedisjoint (simple closed) curves that are realized simultaneously as simple closed geodesics forthe flat metric induced by this one-form.• The statement a) of the theorem is a direct consequence of[46, Prop. A.1].

Proof of Theorem 2.6, caseH(2). Lemma 2.5 almost proves the statement forH(2) except that itdoes not guarantee that both cylinders are simple. We will give here a proof by using [45, Lem.2.1]. Let s be a saddle connection onX that is invariant by the hyperelliptic involutionτ (one canfind such a saddle connection by picking a geodesic segment ofminimal lengths joining a regularWeierstrass point ofX to the unique zero ofω, then takes= s∪ τ(s)). By [45, Lem. 2.1], there existsa simple cylinderC1 that containss. Cut off C1 from X then identify the two geodesic segments onthe boundary of the resulting surface, we obtain a flat torus (X′, ω′) with a marked geodesic segments′.

TRANSLATION SURFACES AND THE CURVE GRAPH IN GENUS TWO 9

We consider (X′, ω′, s′) as a slit torus. By Lemma 2.3, we know that there exists a cylinderC′ in X′

that containss′. The complement ofC′ in X′ is another cylinderC2 whose core curves do not meets′.By constructionC2 is a simple cylinder inX and disjoint fromC1, hence the first assertion follows.

For the second assertion, we observe that any triple of pairwise disjoint simple closed curves dis-connectX into two thrice-holed spheres. If all the curves in this triple are simple closed geodesics(core curves of cylinders), then we get two flat surfaces withgeodesic boundary. SinceX has onlyone singularity, one of the surfaces has no singularities inits interior. But the Euler characteristic ofa thrice-holed sphere is−1, thus we have a contradiction to the Gauss-Bonnet formula.We can thenconclude thatX can not contain three disjoint cylinders.

Proof of Theorem 2.6, caseH(1, 1). By [45, Lem. 2.1], we know that there exists a simple cylinderC0 on (X, ω) that is invariant byτ. Cut off C0 and glue the two boundary components of the resultingsurface, we obtain a surface (X, ω) ∈ H(2) with a marked saddle connection ˆs. Note that ˆs is invariantby the hyperelliptic involution ofX. By Lemma 2.5, we know that there exist two cylindersC1 andC2 on X disjoint from s such thatC1 ∩ C2 = ∅. It follows immediately thatC1 andC2 are actuallycylinders inX and disjoint fromC0, from which we get the desired conclusion.

3. Degenerate cylinders and cylinder graph

3.1. Cylinder and the curve graph. Each cylinder in a translation surface is filled by simple closedgeodesics in the same free homotopy class. The following elementary lemma shows that two (freely)homotopic closed geodesics must belong to the same cylinder.

Lemma 3.1. Let c1 and c2 be two simple closed geodesics in(X, ω) which are freely homotopic. Thenc1 and c2 are contained in the same cylinder.

Proof. Sincec1, c2 are freely homotopic, they are homologous, henceω(c1) = ω(c2). It follows thatc1 andc2 are parallel, thus must be disjoint. The pairc1, c2 cut X into two components, one of whichmust be an annulus denoted byA (see [9, Prop. A.11]). We have a flat metric onA induced by theflat metric ofX. Let θ1, . . . , θk be the cone angles at the singularities inA. Since the boundary ofA isgeodesic, the Gauss-Bonnet formula gives

∑

1≤i≤k

(2π − θi) = 2πχ(A) = 0.

Since any singularity on a translation surface has cone angle at least 4π, the equation above actuallyshows thatA contains no singularities. ThusA is a flat annulus, which must be contained in a cylinderof X. Therefore,c1 andc2 are contained in the same cylinder.

Let S be a fixed topological compact closed surface of genus two. Let C(S) denote the curve graphof S. LetΩT2 be the Abelian differential bundle over the Teichmüller spaceT2. Elements ofΩT2 areequivalence classes of triples (X, ω, f ), whereX is a Riemann surface of genus two,ω is a holomorphicone-form onX, and f is a homeomorphism fromS to X; two triples (X, ω, f ) and (X′, ω′, f ′) areidentified if there exists an isomorphismϕ : X → X′ such thatϕ∗ω′ = ω and f ′−1 ϕ f : S → S isisotopic to idS. The equivalence class of (X, ω, f ) will be denoted by [X, ω, f ].

10 DUC-MANH NGUYEN

Each element [X, ω, f ] of ΩT2 defines naturally a subgraphCcyl(X, ω, f ) of C(S). The verticesof this subgraph are free homotopy classes of the core curvesof all cylinders on the translation sur-face (X, ω). The setC(1)

cyl(X, ω, f ) consists of the edges inC(1)(S) whose both endpoints belong to

C(0)cyl(X, ω, f ).

3.2. Degenerate Cylinders. If C be a cylinder inX that fills X (i.e. C = X), thenC represents anisolated vertex inCcyl(X, ω, f ). This is because the core curve of any other cylinder inX must crossC. So in generalCcyl(X, ω, f ) is not a connected graph. To fix this issue we introduce the notionof degenerate cylinders. Roughly speaking, a degenerate cylinder inX is a union of parallel saddleconnections such that there exist deformations of (X, ω) where this union is freely homotopic to thecore curves of a simple cylinder.

To be more precise, letx0 be a singularity on a translation surface (X, ω). For any pair (r1, r2) ofgeodesic rays emanating fromx0, we will denote the counterclockwise angle fromr1 to r2 byϑ(r1, r2).If s is an oriented saddle connection from a singularityx1 to a singularityx2, then we denote bys+

(resp.s−) the intersection ofswith a neighborhood ofx1 (resp. a neighborhood ofx2). This definitionalso makes sense whenx1 = x2, in which case the orientation ofs is to start ins+ and end ins−.

Definition 3.2 (Degenerate cylinder). We will call the union of two saddle connections s1, s2 in(X, ω) ∈ H(2) ⊔ H(1, 1) a degenerate cylinderif they are both invariant by the hyperelliptic invo-lution, and up to an appropriate choice for the orientationsof s1 and s2, we have

ϑ(s−1 , s+

2) = ϑ(s+1 , s−2) = π.

In Figure 3, we represent the configurations of a degenerate cylinder at the singularities.

π

π

s+1s−2

s−1 s+2

CaseH(2)

π

πs1

s2

CaseH(1, 1)

Figure 3. Configuration of a degenerate cylinder at the singularities.

Remark 3.3.• If (X, ω) ∈ H(2), then a degenerate cylinder is not a simple curve, the zero ofω is its unique

double point.• If (X, ω) ∈ H(1, 1) then the hyperelliptic involutionτ of X permutes the zeros ofω, thus a

saddle connection invariant byτ must connect the two zeros ofω. Therefore a degeneratecylinder must be a simple closed curve.

Examples:Assume that (X, ω) ∈ H(2)⊔H(1, 1) is horizontally periodic, and has a unique (geometric)horizontal cylinderC. If (X, ω) ∈ H(2) then it has 3 horizontal saddle connectionss1, s2, s3, which are

TRANSLATION SURFACES AND THE CURVE GRAPH IN GENUS TWO 11

contained in the boundary ofC (see Figure 4). Note that all of them are invariant by the hyperellipticinvolution. By definitions1 ∪ s2, s2 ∪ s3, s3 ∪ s1 are three degenerate cylinders. Similarly, if (X, ω) ∈H(1, 1), then we have 4 horizontal saddle connections denoted bys1, . . . , s4 (see Figure 4) such thatsi ∪ si+1 is a degenerate cylinder, fori = 1, . . . , 4, with the conventions5 = s1.

s1

s1

s1

s1

s2

s2

s2

s2

s3

s3

s3

s3

s4

s4

C C

ω ∈ H(2) ω ∈ H(1, 1)

Figure 4. Degenerate cylinders on a horizontally periodic surfacewith a unique geo-metric horizontal cylinder.

We will now prove some key properties of degenerate cylinders.

Lemma 3.4. Let s := s1 ∪ s2 be a horizontal degenerate cylinder in(X, ω) ∈ H(2)⊔ H(1, 1). Thenthere exists in a neighborhood of(X, ω) a continuous family of translation surfaces(Xt, ωt), t ∈ [0, ǫ)in the same stratum as(X, ω), with ǫ ∈ R>0, such that

• (X0, ω0) = (X, ω),• for any t∈ (0, ǫ), (Xt, ωt) contains two saddle connections s1,t, s2,t corresponding to s1, s2 and

satisfy the following property: s1,t ∪ s2,t is freely homotopic to the core curves of a simplecylinder Ct in Xt,• as t→ 0, the width of Ct decreases to zero.

Moreover, for all t∈ (0, ǫ), any vertical saddle connection (resp. regular geodesic) in (X, ω) corre-sponds to a vertical saddle connection (resp. regular geodesic) in (Xt, ωt).

Proof. Let us define ahalf cylinder to be the quotient (R × [0, h])/Γ, whereΓ ≃ Z2 ⋉ Z is generatedby t : (x, y) 7→ (x + ℓ, y) and s : (x, y) 7→ (−x, h − y). We will call h andℓ respectively thewidthandcircumferenceof the half disc. We will refer to the projection of (0, 0) as the marked point onits boundary. Equivalently, a half cylinder is a closed discequipped with a flat metric structure withgeodesic boundary and two singularities of angleπ in the interior.

Recall that all Riemann surfaces of genus two are hyperelliptic. Let p : X → CP1 be the hyper-elliptic double cover ofX. There exists a meromorphic quadratic differentialη onCP1 with at mostsimple poles such thatω2

= p∗η. Note thatη has one zero, andk poles, wherek = 5 if ω ∈ H(2), andk = 6 if ω ∈ H(1, 1). LetP0 denote the unique zero ofη, andP1, . . . ,Pk its simple poles. LetY be theflat surface defined byη onCP1. Observe that the cone angle ofY at P0 is 3π if ω ∈ H(2), and 4π ifω ∈ H(1, 1). The cone angle atPi is π, for 1, . . . , k.

Sincesi , i = 1, 2, is invariant byτ, its projection inY is a geodesic segments′i joining P0 to a poleof η. By the definition of degenerate cylinder, one of the angles at P0 specified bys′1 ands′2 is π. LetY be the flat surface obtained by slitting openY alongs′1 ands′2. By construction,Y is a flat disc withk − 2 singularities (of cone angleπ) in its interior, and whose boundary is a geodesic loopc based atP0. Note thatP0 is also a singular point ofY.

Let c denote the boundary ofY, andℓ be the length ofc. Fix anǫ > 0. For anyt ∈ (0, ǫ), let Ct

be the half cylinder of circumference equal toℓ, and width equal tot. We can glueCt to Y such that

12 DUC-MANH NGUYEN

the marked point in the boundary ofCt is identified withP0. Let Y′t denote the resulting flat surface.Observe thatY′t corresponds to a meromorphic differentialη′t onCP1 which has a unique zero atP0

and the same number of simple poles asη. It follows that the orienting double cover of (CP1, η′t ) isan Abelian differential (Xt, ωt) in the same stratum as (X, ω). Remark also that the double cover ofCt

is a simple cylinder of width equal tot. We define (X0, ω0) to be (X, ω). It is now straightforward tocheck that the family(Xt, ωt), t ∈ [0, ǫ) satisfies the properties in the statement of the lemma.

As a by product of Lemma 3.4, we also have

Lemma 3.5. Let s:= s1 ∪ s2 be a degenerate horizontal cylinder in(X, ω) ∈ H(2)⊔H(1, 1).

(i) If (X, ω) ∈ H(2), then there exist a pair of homologous saddle connections r± that cut out aslit torus containing s satisfying the following condition: any vertical leaf crossing r± mustintersect s.

(ii) If (X, ω) ∈ H(1, 1), then eithera) there exist a pair of homologous saddle connections r± that cut out a a slit torus contain-

ing s such that any vertical leaf crossing r± must intersect s, orb) there are two simple cylinders C1,C2 disjoint from s such that any vertical leaf crossing

C1 or C2 must intersect s.

Proof. Let us use the same notations as in the proof of Lemma 3.4. Recall that by slitting openYalong the projections ofs1 and s2, we obtain a flat surfaceY, whose boundary is a geodesic loopcbased atP0. One can construct a new flat surface homeomorphic to the sphereCP1 by “sewing up”c.This operation produces an extra singular point of angleπ at the midpoint ofc.

Let Y′ denote the resulting surface. OnY′, we havek − 1 singularities of cone anglesπ and asingularity atP0 of cone angle 2π if ω ∈ H(2), or 3π if ω ∈ H(1, 1). The loopc corresponds to asegmentc′ onY′ joining P0 to a singularity of angleπ. Let (X′, ω′) be the orienting double cover ofY′,then either (X′, ω′) ∈ H(0, 0), or (X′, ω′) ∈ H(2). In both cases,c′ gives rise to a saddle connections′

invariant by the hyperelliptic involution ofX′. Note that by construction, we can identifyX′ \ s′ withX \ s.

Let ϕ : P → X′ be the embedded parallelogram associated tos′ introduced in Lemma 2.1. Byconstruction,ϕ maps the sides ofP to saddle connections onX′ which do not intersects′ in theirinterior. Thus those saddle connections correspond to somesaddle connections onX. It follows thatϕ(P) ⊂ X′ corresponds to a subsurface ofX containings. The conclusions of the lemma then followfrom a careful inspection on the boundary ofϕ(P).

3.3. Cylinder graph. We now define a new subgraphCcyl(X, ω, f ) of C(S) as follows: the verticesof Ccyl(X, ω, f ) are free homotopy classes of core curves of cylinders, or free homotopy classes ofdegenerate cylinders inX. Elements ofC(1)

cyl(X, ω, f ) are the edges ofC(S) whose both endpoints are

in C(0)cyl(X, ω, f ).

Let dC denote the distance inC(S). Recall that by definition each edge ofC(S) has length equalto one. Leta, b be two simple closed curves onS, and [a], [b] be respectively their free homotopyclasses considered as vertices ofC(S). We have

dC([a], [b]) = minleng(γ), γ path inC(S) from [a] to [b].

TRANSLATION SURFACES AND THE CURVE GRAPH IN GENUS TWO 13

We define a distanced in Ccyl(X, ω, f ) in the same manner, that is, every edge has length equal to one,and given [a], [b] ∈ Ccyl(X, ω, f ),

d([a], [b]) = minleng(γ), γ path inCcyl(X, ω, f ) from [a] to [b].By convention, if there are no paths inCcyl(X, ω, f ) from [a] to [b], then we defined([a], [b]) = ∞.The subgraphCcyl(X, ω, f ) will be the main subject of our investigation in the remaining of this paper.To alleviate the notations, when (X, ω) and a marking mappingf : S → X are fixed, we will writeCcyl andCcyl instead ofCcyl(X, ω, f ) andCcyl(X, ω, f ).

Convention: In the sequel, a “cylinder” could mean a usual geometric cylinder or a degenerate one.We will refer to geometric usual cylinders asnon-degeneratecylinders. The term “a core curve” willhave the usual meaning for non-degenerate cylinder, for a degenerate one it just means the cylinderitself.

3.4. Intersection numbers. Let ι(., .) denote the geometric intersection form on the set of free ho-motopy classes of simple closed curves onS. Let a, b be two simple closed curves inS, and [a], [b]their free homotopy classes respectively. Recall that [a] and [b] are connected by an edge inC(S) ifand only if ι([a], [b]) = 0.

Assume now thata andb are simple closed geodesics in (X, ω). If a andb are parallel, then theydo not have intersection, henceι([a], [b]) = 0. If they are not parallel, then they intersect transversallyat every intersection point. By using the bigon criterion (see [15, Section 1.2.4]), it is not difficult toshow thatι([a], [b]) = #a ∩ b. However, ifa or b is a degenerate cylinder then we must be a littlemore careful since in this casea or b may be not a simple curve (i.e. inH(2)), and their intersectionsare not always transversal.

To deal with this complication, ifa and b are core curves of two cylinders inX (possibly de-generate), we will fix some parametrizationsα : S1 → X for a, and β : S1 → X for b suchthat α andβ are locally homeomorphic onto their images, and the restriction of α (resp. ofβ) toS1 \ α−1(singularities ofX) (resp. toS1 \ β−1(singularities ofX)) is one-to-one.

By an intersectionof a andb, we will mean a pair (t, t′) ∈ S1 × S1 such thatα(t) = β(t′). Thisintersection is said to betransversalif there existǫ > 0 andǫ′ > 0 such thata1 := α((t − ǫ, t + ǫ)) andb1 := β((t′ − ǫ′, t′ + ǫ′)) are two simple arcs inX, a1 intersectsb1 transversally atp = α(t) = β(t′), anda1 andb1 have no other intersections. We denote bya∩ b the set of intersections ofa andb , and bya∩b the subset of transversal intersections.

Lemma 3.6. Let C and D be two cylinders on(X, ω) (both possibly degenerate) that are not parallel.Let c and d be respectively a core curve of C and a core curve of D. We denote by[c] and [d] the freehomotopy classes of c and d respectively. Let c∩d denote the set of transversal intersections of c andd. Then we have

ι([c], [d]) = #c∩d.Since a non-transversal intersection of c and d can only occur at a singularity, it follows in particularthat ι([c], [d]) = #c∩ d if one of c and d is a regular geodesic.

Proof. Let π : ∆ = z ∈ C, |z| < 1 → X denote the universal cover ofX. The pull-backπ∗ω of ω isa holomorphic one-form, which defines a flat metric with cone singularities on∆.

Fix a base pointx for c and a base pointy for d, which are not the singularities ofX. Through anypoint inπ−1(x) (resp. any point inπ−1(y)), there is a unique lift ofc (resp. a unique liftd). Sincec

14 DUC-MANH NGUYEN

andd are not necessarily simple curves,a priori each lift ofc andd may not be a simple arc. But thisactually does not happen.

Claim 3.7.

(i) Each lift of c (resp. of d) is a simple arc in∆.(ii) Two lifts of c (resp. of d) can only meet at at most one point (which is a non-transversal

intersection).(iii) A lift of c and a lift of d can only meet at at most one point.

Proof of the claim.Since the argument for the three assertions are the same, we only give the proof of(iii). Let c0 andd0 be a lift ofc and a lift ofd in ∆ respectively. Let us assume that ˜c0 andd0 intersectat two points. There exists then a discB ⊂ ∆ bounded by a subarcc0 ⊂ c0 and a subarcd0 ⊂ d0. Letp, q be the common endpoints ofc0 andd0, andα andβ be respectively the interior angles ofB at pandq. Sincec0 andd0 are geodesic segments for the flat metric on∆, we haveα > 0 andβ > 0 (α = 0or β = 0 means thatc andd are parallel).

Let p1, . . . , pr be the points in∂B that correspond to the zeros ofπ∗ω and different fromp, q. Letθi be the interior angle ofB at pi. By definition of cylinders, we haveθi ≥ π, for all i = 1, . . . , r. Letx1, . . . , xs be the zeros ofπ∗ω in int(B), andθi be the angles atxi. The Gauss-Bonnet formula gives(see for instance [51, Prop. 1])

s∑

i=1

(2π − θi) +r∑

i=1

(π − θi) + 2π − (α + β) = 2πχ(B) = 2π.

Sinceα + β > 0, π − θi ≤ 0, and 2π − θi < 0, we see that the equality above cannot be realized.Therefore,B cannot exist, which means that ˜c0 andd0 can only meet at at most one point.

Since non-transversal intersections ofc andd can only occur at the singularities ofX (zeros ofω),we can deformc andd slightly in a neighborhood of each zero ofω to get simple closed curvesc′ andd′ in the same free homotopy classes asc andd respectively such that #c∩d = #c′ ∩ d′. Claim 3.7then implies that any lift ofc′ in ∆ intersects a lift ofd′ at at most one point and all the intersectionsare transversal. It follows from the bigon criterion (seee.g. [15, Prop. 1.7]) that

ι([c], [d]) = #c′ ∩ d′ = #c∩d.

The lemma is then proved.

Remark 3.8.

• If C and D are not parallel, we can assume that C is horizontal and D is vertical. In the caseboth C and D are degenerate, to compute their intersection number, one can use Lemma 3.4to get a deformation(Xt, ωt) of (X, ω) in which C corresponds to simple (horizontal) cylinderCt. In Xt, D corresponds to a vertical cylinder Dt. Consequently, c is freely homotopic to aregular horizontal geodesic ct in Xt, while d is freely homotopic to a core curve dt of Dt. Itfollows from Lemma 3.6 thatι([c], [d]) = ι([ct], [dt]) = #ct ∩ dt.• It may happen that two degenerate cylinders in the same direction have a positive intersection

number.

TRANSLATION SURFACES AND THE CURVE GRAPH IN GENUS TWO 15

4. Reducing numbers of intersection

In what follows, given two cylindersC,D in X, by ι(C,D) we will mean the geometric intersectionnumberι([c], [d]), wherec andd are some core curves ofC andD respectively. Our first goal is toestimate the distance inCcyl by using intersection numbers.

Theorem 4.1. There exist two positive constants K1,K2 such that for any[X, ω, f ] ∈ ΩT2, and anycylinders C and D in X (both possibly degenerate) consideredas vertices ofCcyl(X, ω, f ), we have

(1) d(C,D) ≤ K1ι(C,D) + K2.

As a direct consequence of inequality (1), we get

Corollary 4.2. The subgraphCcyl(X, ω, f ) is connected.

4.1. Reducing to simple cylinders. In what follows, we will fix a point [X, ω, f ] ∈ ΩT2, and bycylinders inX we include degenerate ones. Our first step is to reduce the problem to the caseC andDare simple cylinders.

Lemma 4.3. Let C be horizontal cylinder that does not fill X,i.e. C , X, and D be a verticalcylinder. Assume thatι(C,D) > 0. Then there exists a simple cylinder C′ such thatd(C,C′) ≤ 1 andι(C′,D) ≤ ι(C,D).

Proof. We first consider the caseC is non-degenerate. Letc be a core curve ofC andd a core curve ofD. Sincec is a regular simple closed geodesic, by Lemma 3.6, we haveι(C,D) = #c∩d. Obviously,we only need to consider the caseC is not simple.

If (X, ω) ∈ H(2), then the complement ofC is a simple cylinderC′ whose boundary is a pairhomologous saddle connections contained in the boundary ofC. In particularC′ is also horizontal,and we haveι(C,C′) = 0, henced(C,C′) = 1. Any timed crossesC′, it must crossC before returningto C′. Therefore, we haveι(C′,D) ≤ ι(C,D).

If (X, ω) ∈ H(1, 1) then the complement ofC is either: (a) horizontal simple cylinder, (b) twodisjoint horizontal simple cylinders, or (c) a torus with a horizontal slit. In case (a) and case (b),the boundaries of the horizontal cylinders in the complement are contained in the boundary ofC.Therefore, it suffices to choose one of them to beC′. In case (c), let (X′, ω′, s′) be the slit torus whichis the complement ofC. Note that the slits′ corresponds to a pair of homologous saddle connectionsin the boundary ofC. By Lemma 2.3 we know thatX′ contains a simple cylinderC′ disjoint fromthe slit s′ such that any vertical line crossingC′ must crosss′. SinceC′ is disjoint fromC we haved(C,C′) = 1. Any timed crossesC′, it must cross the slits′ and henceC. Therefore, we also haveι(C′,D) ≤ ι(C,D).

We now turn to the caseC is degenerate. If (X, ω) ∈ H(2), from Lemma 3.5, we know thatC iscontained in a slit torus cut out by a pair of homologous saddle connectionsr± such that every verticalleaf crossingr± intersectsC. Since (X, ω) ∈ H(2), the complement of the slit torus is a simple cylinderC′ bounded byr±. Clearly, we haved(C,C′) = 1. If the core curves ofD are regular geodesics (thatis D is non-degenerate), then we can immediately conclude thatι(C′,D) ≤ ι(C,D). In caseD isdegenerate, we consider the deformations(Xt, ωt), t ∈ [0, ǫ) of (X, ω) given by Lemma 3.4. Fort ∈ (0, ǫ), in (Xt, ωt), D becomes a simple cylinderDt, while the cylindersC andC′ persist and havethe same properties. Sinceι(C′,D) = ι(C′,Dt) andι(C,D) = ι(C,Dt), we also getι(C′,D) ≤ ι(C,D).

16 DUC-MANH NGUYEN

The case (X, ω) ∈ H(1, 1) also follows from similar arguments.

Lemma 4.4. Assume that C is a horizontal cylinder that fills X, and D is a vertical cylinder. Thenthere exists a simple cylinder C′ such that

d(C′,C) = 2,ι(C′,D) ≤ ι(C,D).

Proof. Let c be a core curve ofC. If (X, ω) ∈ H(2) then the complement ofC is the union of threehorizontal saddle connectionss1, s2, s3, all are invariant by the hyperelliptic involution. Remarkthatthe union of any two of these saddle connections is a degenerate cylinder. One can easily find atransverse simple cylinderC′ containings1, disjoint from the unions2∪ s3, whose core curves crossconce. Furthermore, we can chooseC′ such that the horizontal component of its core curves has lengthsmaller than the length ofc. Clearly, we haved(C,C′) = 2. Since any vertical geodesic crossingC′

crosses alsoC, we haveι(C′,D) ≤ ι(C,D). Thus the lemma is proved for this case.The case (X, ω) ∈ H(1, 1) follows from the same arguments.

In what follows, a geodesic line onX that does not contain any singularity is calledregular.

Lemma 4.5. Let C be a horizontal cylinder and D be a vertical cylinder in X. If there exists a regularvertical leaf which does not cross C thend(C,D) ≤ 2.

Proof. Obviously we only need to consider the caseι(C,D) > 0. Assume that there is a regularvertical closed geodesic that does not intersectC, then there exists another vertical cylinderD′ whichis disjoint from bothC andD. Consequently, we haved(C,D) = 2.

Assume now that there is an infinite regular vertical leaf that does not intersectC. The closureof this leaf is a subsurfaceX′ of X bounded by some vertical saddle connections. Lets be a saddleconnection in the boundary ofX′. Note thatsandτ(s) are homologous. Thus they decomposeX intotwo subsurfacesX1 andX2 both invariant byτ. SinceC is invariant byτ, it must be contained in oneof the subsurfaces, sayX1. Sincesandτ(s) are vertical, the core curves ofD cannot crosssandτ(s),which means thatD is also contained in one subsurface. Since we have assumed that ι(C,D) > 0, Dmust be contained inX1.

The subsurfaceX2 must be either a slit torus, or a surface inH(2) with a marked saddle connection.Actually, the latter case does not occur because it would imply that X1 is a vertical simple cylindercontaining bothC andD, which is impossible. Now, by Lemma 2.3, one can find in the torus X2 asimple cylinderC′ that does not meet the slit. SinceC′ corresponds to a simple cylinder ofX whichis disjoint from bothC andD, and we haved(C,D) = 2. The lemma is then proved.

From the Lemmas 4.3, 4.4, we know that ifC is not simple then there exists a simple cylinderC′

such thatd(C,C′) ≤ 2 andι(C′,D) ≤ ι(C,D). Consequently, we can find simple cylindersC′,D′ suchthat

d(D,D′) ≤ 2,d(C,C′) ≤ 2,ι(C′,D′) ≤ ι(C,D).

It follows in particular thatd(C,D) ≤ d(C′,D′) + 4. Therefore we only need to prove (1) for the caseC andD are simple cylinders. Moreover, by Lemma 4.5, we can furtherassume that all the leaves ofthe foliation in the direction ofD intersectC. Thus, Theorem 4.1 is a consequence of the following

TRANSLATION SURFACES AND THE CURVE GRAPH IN GENUS TWO 17

Proposition 4.6. Let C and D be two simple cylinders such that all the leaves of the foliation in thedirection of D intersectC. Then there always exists a simple cylinder C′ such that

(2) d(C′,C) ≤ 3 and ι(C′,D) < ι(C,D).

To prove this proposition we will make use of the representation of translation surfaces as polygonsin R2. In Section A, we give a uniform construction from symmetricpolygons of translation surfacesin genus two satisfying the hypothesis of Proposition 4.6.

4.2. Proof of Proposition 4.6, CaseH(2).

Proof. By using GL+(2,R), we can assume thatC is a horizontal cylinder, andD is vertical. FromProposition A.1(i), we can construct (X, ω) from a symmetric polygonP := (P0 . . .P3Q0 . . .Q3) inR2. Note that by construction, the hyperelliptic involution of X lifts to the central symmetry fixing themidpoint ofP0Q0.

P0

P1

P2

P3

Q0

Q1

Q2

Q3

X1 X2

Y

Casex2 ≤ y < x3

P0

P1

P2

P3

Q0

Q1

Q2

Q3

X1

Y X2

Casex1 ≤ y < x2

P0

P1

P2

P3

Q0

Q1

Q2

Q3Z

Y

X1

X2

Case 0< y < x1

Figure 5. Finding simple cylinders having less intersections withD, caseH(2).

Let X1,X2, andY be respectively the vertical projections ofP1,P2, andQ0 onP0P3. Let x1, x2, x3, ybe respectively the lengths ofP0X1,P0X2,P0P3,P0Y. Clearly, we have 0≤ x1 ≤ x2 ≤ x3 and0 ≤ y ≤ x3. Remark that by cutting and gluing, the casesy = 0 (Y ≡ P0) andy = x3 (Y ≡ P3) areequivalent. Therefore we can always suppose 0< y ≤ x3.

By symmetry, we can assume that|P1X1| ≥ |P2X2| (see Figure 5). Observe that the union of theprojections of (P0P1P2) and (Q0Q1Q2) in X is a cylinderE which is disjoint fromC. Similarly, theunion of the projections of (P2P3Q0) and (Q2Q3P0) is also a cylinderF in X, which is disjoint fromE. Observe that by assumption,E is always a simple cylinder, butF can be a degenerate one (that iswhen bothP2P3 andP3Q0 are vertical). Note that we haved(C,E) = 1 andd(C, F) = 2.

Let d be a core curve ofD andd be the pre-image ofd in P. Remark thatd is a (finite) union ofvertical segments with endpoints in the boundary ofP and none of the vertices ofP is contained ind.We first consider the generic case, where none of the sides ofP is vertical. By assumption, we have

0 < x1 < x2 < x3 and 0< y < x3.

18 DUC-MANH NGUYEN

We have three possibilities:

(a) x2 ≤ y < x. We observe that if a vertical line intersectsP0P2 or P2Q0 then it must intersectP0X2 or X2Y respectively. Thus, we have

#d∩ P0P3 ≥ #d ∩ P0P2 + #d∩ P2Q0.It follows that at least one of the following inequalities istrue

[#d ∩ P0P2 < #d∩ P0P3 ⇒ ι(E,D) < ι(C,D),#d ∩ P2Q0 < #d∩ P0P3 ⇒ ι(F,D) < ι(C,D).

Therefore, in this case, we can chooseC′ to be eitherE or F.

(b) x1 ≤ y < x2. Remark that in this case the parallelogram (P0P1Q0Q1) is contained inP,thus it projects to a simple cylinderG in X, which is disjoint fromF. In particular, we haved(G,C) ≤ 3. We now observe that

#d∩ X1X2 = #d∩ P1Q0 + #d ∩ P2Q0 ≤ #d∩ P0P3.Therefore, at least one of the following inequalities is true ι(F,D) < ι(C,D) or ι(G,D) <ι(C,D). Hence we can chooseC′ to be eitherF or G.

(c) 0< y < x1. We will show that in this caseι(G,D) < ι(C,D). LetZ be the vertical projection ofP0 to Q0Q3. We choose a core curved of D which is contained in theǫ-neighborhood of theleft boundary ofD, with ǫ > 0 small. The left boundary ofD is a vertical saddle connection,thus it contains (the projection of) one of the following segmentsP0Z,P1X1,P2X2. It followsthat, d contains a vertical segmentd0 which is ǫ-close to one ofP0Z,P1X1,P2X2 from theright. Observe thatd0 always intersectsP0P3, but whenǫ is chosen to be small enough,d0

does not intersectP1Q0. Since any vertical segment inP intersectingP1Q0 must intersectYX1 ⊂ P0P3, it follows thatι(G,D) < ι(C,D), and we can chooseC′ to beG.

It remains to show that the same arguments work in the degenerating situations, that is when oneof the sides ofP is vertical. First, let us suppose thatP2P3 is vertical, (i.e. x2 = x3).

• If y = x3 thenF becomes a degenerate cylinder. ClearlyF andD are disjoint since they areboth vertical. Therefored(C,D) ≤ d(C, F) + 1 ≤ 3, hence we can chooseC′ to beD.• If 0 < y < x3 then Case (a) and Case (b) then follow from the same arguments. For Case

(c), we observe that the left boundary ofD is not invariant by the hyperelliptic involution, andP2P3 projects to an invariant saddle connection. Therefored0 is eitherǫ-close toP0Z or P1X1.Hence we can use the same argument to conclude thatι(G,D) < ι(C,D) and we can chooseC′ to beG.

Other degenerations are easy to deal with in similar manner,details are left for the reader.

4.3. Proof of Proposition 4.6, CaseH(1, 1).

Proof. Using the notations in Proposition A.1(ii), we know that (X, ω) is obtained from a decagonP := (P0 . . .P4Q0 . . .Q4) ⊂ R2. Our arguments depend on the properties of this decagon. We havethree different models forP (see Figure 6): (I) both int(P0P2) and int(P2P4) are contained in int(P), (II)only one of int(P0P2) and int(P2P4) is contained in int(P), and (III) none of int(P0P2) and int(P2P4)is contained in int(P).

TRANSLATION SURFACES AND THE CURVE GRAPH IN GENUS TWO 19

Let X1,X2,X3, andY be respectively the vertical projections ofP1,P2,P3, andQ0 on P0P4. Thelengths ofP0Xi , P0P4, andP0Y are denoted byxi, x4, andy respectively. As in the previous case, wehave 0≤ xi ≤ xi+1, i = 1, 2, 3, and 0< y ≤ x4. Let d be a core curve ofD, andd its pre-image inP.

P0 P4

Q4 Q0

P1

P2

P3

Q3

Q2

Q1

Model I

P0 P4

Q4 Q0

P1

P2

P3

Q3

Q2

Q1

X1 X2 X3 Y

Model II

P0 P4

Q4 Q0

P1

P2

P3

Q3

Q2

Q1

X1 X2 X3 Y

Model III

Figure 6. Constructing (X, ω) from a decagon.

4.3.1. Model I. In this model, the sets (P0P1P2) ∪ (Q0Q1Q2) and (P2P3P4) ∪ (Q2Q3Q4) projectto two disjoint simple cylinders inX which will be denoted byE and F respectively. Note thatd(C,E) = d(C, F) = 1. Clearly, we have

#d∩ P0P4 = #d∩ P0P2 + #d∩ P2P4 ⇒ ι(C,D) = ι(E,D) + ι(F,D).

Therefore, we can pickC′ to beE or F.

4.3.2. Model II. By symmetry, we only need to consider the case int(P0P2) ⊂ int(P), and int(P2P4) 1int(P). Let E be the simple cylinder onX which is the projection of (P0P1P2) ∪ (Q0Q1Q2). Let F bethe cylinder which is the projection of (P3P4Q0) ∪ (Q3Q4P0). We haved(C,E) = 1 andd(C, F) = 2.

We first consider the generic situation, that is 0< xi < xi+1, i = 1, 2, 3, and 0< y < x4. Note thatin this situationF is a simple cylinder. We have three cases: (a)x2 ≤ y < x4, (b) x1 ≤ y < x2, (c)0 < y < x1. In all of these cases, one can find a simple cylinder having the desired property by thesame arguments as the case (X, ω) ∈ H(2).

Consider now the degenerating situations: (1)P0P1 is vertical⇔ x1 = 0, (2) P1P2 is vertical⇔ x1 = x2, (3) P2P3 is vertical⇔ x2 = x3, (4) P3P4 is vertical⇔ x3 = x4, (5) Y ≡ P4 ⇔ y = x4.If (4) or (5) does not occur thenF is always a simple cylinder, hence the arguments above apply. If(4) and (5) hold thenF is a vertical degenerate cylinder. SinceF must be disjoint fromD, we haved(C,D) ≤ 3. Therefore, we can chooseC′ to beD.

4.3.3. Model III. In this caseP2 must be the highest point ofP, andP1P3 must be contained inP.Consequently, the union (P1P2P3) ∪ (Q1Q2Q3) projects to a simple cylinderE in X. Let F denotethe cylinder inX which is the projection of (P3P4Q0) ∪ (Q3Q4P0). Remark thatd(C,E) = 1 andd(C, F) = 2. It is not difficult to see that the same arguments as the previous cases alsoallow us to getthe desired conclusion.

20 DUC-MANH NGUYEN

4.4. Proof of Theorem 4.1.

Proof. By the Lemmas 4.3, 4.4, we know that there exist two simple cylindersC′ andD′ such thatι(C′,D′) ≤ ι(C,D)d(C,D) ≤ d(C′,D′) + 4.

It follows from Lemma 4.5 and Proposition 4.6 thatd(C′,D′) ≤ 3ι(C′,D′) + 2. Therefore

d(C,D) ≤ 3ι(C,D) + 6.

5. Infinite diameter

In this section we prove

Proposition 5.1. For any(X, ω) ∈ H(2)⊔H(1, 1), the diameter ofCcyl(X, ω, f ) is infinite.

The geometry of the curve complex is closely related to the Teichmüller spaceT (S). Recall thatgiven a simple closed curveγ onS, for anyx ∈ T (S) the extremal length Extx(γ) of γ is defined to be

Extx(γ) = suph|γ∗|2h,

whereh ranges over the set of Riemannian metrics of area one in the conformal class ofx, and|γ∗|his the length of the shortest curve (with respect toh) in the homotopy class ofγ. Alternatively, onecan define Extx(γ) to be the inverse of the largest modulus of an annulus homotopic toγ on S. Thereis a natural coarse mappingΦ from T (S) to C(S) defined as follows, we assign to eachx ∈ T (S) acurve of minimalx-extremal length onS. It is a well known fact (see [35, Lem. 2.4]) that there is auniversal constantc depending only the topology ofS, such that the diameter of the subset ofC(S)consisting of simple curves having minimalx-extremal length is at mostc for any x ∈ T (S).

Teichmüller geodesics inT (S) throughx are the projections of the linesat ·q, whereq is a holomor-phic quadratic differential onS equipped with the conformal structurex, andat =

(et 00 e−t

), t ∈ R. It is

proven in [35] that ifLq : R → T (S) is a Teichmüller geodesic, thenΦ(Lq(R)) is an un-parametrizedquasi-geodesic inC(S). It may happen that this quasi-geodesic has finite diameter.

The curve graphC(S) has infinite diameter (see [35]). In [29], Klarreich shows that the boundaryat infinity ∂∞C(S) of C(S) can be identified with the space of topological minimal foliationsFmin(S)on S. Recall that a foliation onS is minimal if it has no leaf which is a simple closed curve, herewe consider foliations up to isotopy and Whitehead moves. A characterization of sequences of curvesconverging to a foliation in∂∞C(S) is given by Hamenstädt [18]. It follows from this result that ifthe vertical ofq are minimal thenΦ Lq([0,∞)) is a quasigeodesic of infinite diameter inC(S) (see[19, 20]).

Recall that a geometric (non-degenerate) cylinder on a translation surface is modeled byR ×(0, h)/((x, y) ∼ (x + c, y)), wherec > 0 is its circumference andh is its width. In [53], develop-ing Smillie’s ideas in [48], Vorobets showed the following

Theorem 5.2(Smillie-Vorobets). Given any stratumH(κ) of translation surface, there exists a con-stant K> 0 depending onκ such that, on every translation surface of area one inH(κ), one can finda geometric cylinder of width bounded below by K.

TRANSLATION SURFACES AND THE CURVE GRAPH IN GENUS TWO 21

Proposition 5.1 is an easy consequence of this result and theresults of Klarreich and Hamenstädt.

Proof of Proposition 5.1.Using the action of GL+(2,R), we can always assume that Area(X, ω) = 1and the vertical foliation of (X, ω) is minimal. LetL : R→ T (S) be the Teichmüller geodesic definedby q = ω2. By the results of Klarreich and Hamenstädt, the quasi-geodesicΦ L(R>0) has infinitediameter.

Denote bydC the distance inC(S), and byd the distance inCcyl(X, ω, f ). For any pair (α, β) inCcyl(X, ω, f ), we havedC(α, β) ≤ d(α, β).

For eacht ∈ R, let (Xt, ωt) := at · (X, ω). Given anyR ∈ R>0 there existt1, t2 ∈ (0,+∞) such thatdC(Φ L(t1),Φ L(t2)) ≥ R. Let αi := Φ L(ti). By Theorem 5.2 we know that there is a geometriccylinderCi of width bounded below byK in (Xti , ωti ). Let βi be a core curve ofCi.

The extremal length ofαi in Xi is bounded by a universal constante0(S) (see e.g [43, Lem. 2.1]).Thus by definition, the length of the shortest curveα∗i in the homotopy class ofαi with respect toωtiis bounded bye0(S). Since the width ofCi is at leastK, we have #α∗i ∩ βi ≤ e0(S)/K, which impliesthat ι([αi ], [βi ]) ≤ e0(S)/K.

It is well know that the distance inC(S) is bounded by a linear function of the intersection number(see for instance [35, Lem. 2.1] or [5, Lemma 1.1]). Thus there is a constantM depending only onSsuch thatdC([αi ], [βi ]) ≤ M. Therefore, we have

dC([β1], [β2]) ≥ dC([α1], [α2]) − dC([α1], [β1]) − dC([α2], [β2]) ≥ R− 2M

Sinced(C1,C2) = d([β1], [β2]) ≥ dC([β1], [β2]), the proposition follows.

6. Automorphisms of the cylinder graph

Let Aff+(X, ω) denote the group of affine automorphisms of (X, ω). Recall that elements of Aff+(X, ω)are orientation preserving homeomorphisms ofX that preserve the zero set ofω, and are given byaffine maps in local charts of the flat metric out side of this set (see [27, 37]). Remark that the dif-ferential of such a map (in local chart associated to the flat metric) is a constant matrix in SL(2,R).Thus we have a group homomorphismD : Aff+(X, ω) → SL(2,R) which associates to each elementof Aff+(X, ω) its differential (derivative). The image ofD in SL(2,R) is called the Veech group of(X, ω) and usually denoted by SL(X, ω). Note that the kernel ofD is contained in the group Aut(X) ofautomorphisms ofX, thus must be finite. The group SL(X, ω) can also be viewed as the stabilizer of(X, ω) for the action of SL(2,R).

Given a point [X, ω, f ] ∈ ΩT2, via the markingf : S → X, one can identify Aff+(X, ω) with asubgroup of the Mapping Class Group Mod(S) of S (see [37, Section 5]). An element of Mod(S)induces naturally an automorphism of the curve graphC(S). It is a well known fact every automor-phism ofC(S) arises from an element of Mod(S) ([26, 33]). Since an affine homeomorphism mapscylinders into cylinders, and saddle connections into saddle connections, it is clear that any element ofAff+(X, ω) induces an automorphism of the subgraphCcyl(X, ω, f ). The aim of this section is to show

Proposition 6.1. Let φ be an element ofMod(S) which preserves the subgraphCcyl(X, ω, f ), thatis φ(Ccyl(X, ω, f )) ⊂ Ccyl(X, ω, f ). Thenφ is induced by an affine automorphism inAff+(X, ω). Inparticular, φ realizes an automorphism ofCcyl(X, ω, f ).

Remark 6.2. Proposition 6.1 is equivalent to the following statement: if ψ : X → X is a homeomor-phism satisfying the condition: for any regular simple closed geodesic or degenerate cylinder c,ψ(c)

22 DUC-MANH NGUYEN

is freely homotopic to the core curves of a cylinder (possibly degenerate) on X, thenψ is isotopic toan affine automorphism of(X, ω).

The proof of this proposition essentially follows from the arguments of [14, Lemma 22]. Beforegetting into the proof, let us recall some basic notions of Thurston’s compactification of the Teich-müller space. LetMF (S) denote the space ofmeasured foliationsonS. The space ofprojective mea-sured foliationsdenoted byPMF (S) is naturally the quotient ofMF (S) by R∗+. Thurston showedthatPMF (S) can be identified with the boundary ofT (S). A foliation is minimalif none of its leavesis a closed curve. A (measured) foliation isuniquely ergodicif it is minimal and there exists a uniquetransverse measure up to scalar multiplication.

The set of (free homotopy classes of) simple closed curves inS (that is the vertex set ofC(S)) is nat-urally embedded inMF (S) with the transverse measure being the counting measure of intersections.The geometric intersection numberι(., .) defined on the set of pairs of simple closed curves extends toacontinuous symmetric functionι :MF (S)×MF (S)→ [0,+∞) which satisfiesι(aλ, bµ) = abι(λ, µ),for all a, b ∈ [0,+∞) andλ, µ ∈ MF (S). It has been shown by Thurston that the set

(0,+∞) · α, α is a simple closed curveis dense inMF (S).

Two measured foliations aretopologically equivalentif the corresponding topological foliationsare the same up to isotopy and Whitehead move. The following result was proved in [47]

Proposition 6.3. If λ is a minimal measured foliation, andι(λ, µ) = 0, thenλ andµ are topologicallyequivalent.

Measured foliations are a special case of more general objects calledgeodesic currentswhich wereintroduced by Bonahon (see [3, 4]). We refer to [14] for an introduction to this concept with moredetails. While the space of measure foliations is the completion of the set ofsimple closed curve, thespace of geodesic currents, denoted byC (S), can be viewed as the completion of closed curves onS.In particular, we have a continuous extension of the intersection number functionι to C (S) × C (S).A characterization of measured foliations in the space of current geodesics was given Bonahon in [3,Prop. 4.8]:

Proposition 6.4(Bonahon). MF (S) is exactly the set of geodesic current with zero self-intersection,that is

MF (S) = λ ∈ C (S), ι(λ, λ) = 0.Another important feature of geodesic currents we will needis the following

Proposition 6.5 (Bonahon [4], Prop. 4). Let α be a geodesic current with the following property:every geodesic inS transversely meets another geodesic in the support ofα. Then the setβ ∈ C (S)such thatι(α, β) ≤ 1 is a compact inC (S).

Remark that ifλ is a minimal foliation, then the corresponding geodesic current satisfies the hy-pothesis of Proposition 6.5.

Every holomorphic one-form (X, ω) (or more generally every holomorphic quadratic differential)defines naturally two measured foliations onX. The leaves of these foliations are respectively verticaland horizontal geodesic lines with the transverse measuresgiven by|Reω| and|Imω|. It is also a well-known fact that, ifλ andµ are two uniquely ergodic measured foliations jointly filling upS, that is for

TRANSLATION SURFACES AND THE CURVE GRAPH IN GENUS TWO 23

anyν ∈ MF (S), we haveι(ν, λ) + ι(ν, µ) > 0, then there is a unique Teichmüller geodesicg that joins[λ] and [µ], where [λ] and [µ] are the projections ofλ andµ in PMF (S). As a consequence, assumethat (X1, ω1) and (X2, ω2) are two holomorphic one-forms both satisfy the following condition: thevertical foliation ofωi is topologically equivalent toλ, and the horizontal foliation is topologicallyequivalent toµ. Then there exists a diagonal matrixA =

(et 00 es

)∈ GL+(2,R) such that (X2, ω2) =

A · (X1, ω1).

Proof of Proposition 6.1.

Proof. By definition,φ·[X, ω, f ] = [X, ω, f φ−1]. Equivalently, we can writeφ·[X, ω, f ] = [X′, ω′, f ′],where f ′ : S → X′ satisfies the following condition: there exists an isomorphism φ : X′ → X suchthat φ∗ω = ω′, and f φ−1 is isotopic toφ f ′. Using this identification, we have

Ccyl(X′, ω′, f ′) = φ(Ccyl(X, ω, f )).

Thus, by assumption, we haveCcyl(X′, ω′, f ′) ⊂ Ccyl(X, ω, f ).Via the mapsf : S→ X, f ′ : S→ X′, for any directionθ ∈ RP1, we denote byνθ andν′θ the mea-

sured foliations onS corresponding to the vertical foliations defined byeıθω andeıθω′ respectively.The leaves ofνθ andν′θ are geodesic lines in the direction of±(π/2 − θ). Observe that ifθk is asequence of angles converging toθ, thenνθk converges toνθ, andν′θk converges toν′θ inMF (S).

It follows from a classical result of Kerckhoff-Masur-Smillie [28] that for almost all directionsθ ∈ RP1, νθ (resp.ν′θ) is uniquely ergodic. Set

UE(ω) := [νθ] ∈ PMF (S), νθ is uniquely ergodic,θ ∈ RP1 ⊂ PMF (S).

We defineUE(ω′) in the same manner.We will show thatUE(ω′) ⊂ UE(ω). Letθ be a direction such thatν′θ is uniquely ergodic. Without

loss of generality, we can assume that Area(X) = 1. For anyt ∈ R, set

(X′tθ, ω′t

θ) :=(

et 00 e−t

)· (X′, eiθω′).

It follows from Theorem 5.2 that there exists a constantR > 0 such that for anyt ∈ R, X′tθ has a

cylinderC′t with circumference bounded byR. Let c′t be a core curve ofC′t , and consider the sequencec′kk∈N. By definition, the length ofc′k with respect toω′k

θ, denoted byℓω′kθ (c′k), is bounded byR. Thus

we haveι(ekν′θ, c′k) = ekι(ν′θ, c′k) ≤ ℓω′kθ (c

′k) ≤ R.

It follows thatlim

k→+∞ι(ν′θ, c′k) = 0.

By Proposition 6.5, up to extracting a subsequence, we can assume thatc′k converges to a geodesiccurrentµ′ ∈ C (S). Sincec′k has zero self-intersection, it follows thatι(µ′, µ′) = 0, henceµ′ ∈ MF (S)by Proposition 6.4. By continuity ofι we haveι(ν′θ, µ′) = 0. Sinceν′θ is uniquely ergodic (so inparticular, it is minimal), it follows from Proposition 6.3thatµ′ andν′θ are topologically equivalent.Henceµ′ is also uniquely ergodic.

By definition, c′kk∈N are vertices ofCcyl(X′, ω′, f ′). By assumption, we haveCcyl(X′, ω′, f ′) ⊂Ccyl(X, ω, f ). Therefore,c′kk∈N are also vertices ofCcyl(X, ω, f ), which means thatc′k is freely ho-motopic to either a simple closed geodesic, or a degenerate cylinder in X. In particular, we see that

24 DUC-MANH NGUYEN

eachc′k has a well defined directionθk ∈ RP1 with respect toω. Again, by extracting a subsequence,

we can assume thatθk converges toθ. Thus,νθk converges toνθ. Since we haveι(νθk, c′k) = 0,

by continuity, it follows thatι(νθ, µ′) = 0. Sinceµ′ is uniquely ergodic, so isνθ, and we have[ν′θ] = [µ′] = [νθ] ∈ PMF (S). We can then conclude thatUE(ω′) ⊂ UE(ω).

Now pick a pair of projective uniquely ergodic measured foliations ([λ], [µ]) in UE(ω′) ⊂ UE(ω)that jointly fill up S. There exist two matricesM andM′ such that the vertical and horizontal foliationsof M · [X, ω, f ] (resp. ofM′ · [X′, ω′, f ′]) are topologically equivalent toλ andµ respectively. Sincethere is a unique Teichmüller geodesic joining [λ] and [µ], there must exist a diagonal matrixA ∈GL+(2,R) such thatM′ · [X′, ω′, f ′] = AM · [X, ω, f ], which implies thatφ is represented by an affineautomorphism of (X, ω).

Remark 6.6. This proof actually works for translation surfaces in any genus withCcyl replaced bythe subgraph consisting of non-degenerate cylinders.

7. Hyperbolicity

A translation surface (X, ω) is said to becompletely periodic(in the sense of Calta) if the directionof any non-degenerate cylinder inX is periodic, which means that whenever we find a simple closedgeodesic onX, the surface decomposes as union of (finitely many) cylinders in the same direction(see [10, 11]). It stems out from [10] and [42] that, inH(2), a surface is completely periodic ifand only if it is a Veech surface. InH(1, 1), a surface is completely periodic if and only if it is aneigenform for a real multiplication. In particular, there are completely periodic surfaces inH(1, 1)that are not Veech surfaces.

Let us denote byED, whereD is a natural number such thatD ≡ 0 or 1 mod 4, the locus ofeigenforms for the real multiplication by the quadratic order OD in ΩM2. EachED is a 3 dimen-sional irreducible (algebraic) subvariety ofΩM2 which is invariant by the SL(2,R)-action. The set ofeigenforms inΩM2 is then (see [42])

E =⋃

D≡0,1 mod 4

ED.

Even though complete periodicity is initially defined for directions of non-degenerate cylinders, it isnot difficult to show that in the case of genus two, this property actually implies the periodicity fordirections of degenerate cylinders (see Lemma B.1). Alternatively, one can also use the argumentin [55] to get the same result in more general contexts (see [56]). In what follows, by acompletelyperiodicsurface we will mean a surface for which the direction of any cylinder (degenerate or not) isperiodic. By Lemma B.1, this apparently new definition agrees with the usual one by Calta. Our goalin this section is to show

Theorem 7.1. If (X, ω) ∈ H(2)⊔ H(1, 1) is completely periodic thenCcyl(X, ω, f ) is Gromov hyper-bolic.

To prove this theorem, we will use the following hyperbolicity criterion by Masur-Schleimer [36,Theorem 3.13] (see also [7, Prop. 3.1], and [17]), and followBowditch’s approach in [5].

Theorem 7.2(Masur-Schleimer). Suppose thatX is a graph with all edge lengths equal to one. ThenX is Gromov hyperbolic if there is a constant M≥ 0, and for all unordered pair of vertices x, y inX0,there is a connected subgraph gx,y containing x and y with the following properties

TRANSLATION SURFACES AND THE CURVE GRAPH IN GENUS TWO 25

• (Local) If dX(x, y) ≤ 1 then gx,y has diameter at most M,• (Slim triangle) For any x, y, z ∈ X0, the subgraph gx,y is contained in the M-neighborhood of

gx,z∪ gz,y.

Let us fix [X, ω, f ] ∈ ΩT2, where (X, ω) ∈ E and Area(X, ω) = 1. We will write Ccyl insteadof Ccyl(X, ω, f ). We know from Corollary 4.2 thatCcyl is connected, and by definition the edges ofCcyl have length equal to one. LetK be the constant in Theorem 5.2, andC be a cylinder of widthbounded below byK in X. Note that the circumference ofC is bounded above by 1/K. Recall thatfrom Theorem 4.1, we know that there are two constantsK1,K2 such that for any pair of cylindersC,D in X, we have

d(C,D) ≤ K1ι(C,D) + K2

whered is the distance inCcyl(X, ω, f ), andι(C,D) is the number of intersections of a core curve ofCand a core curve ofD.

7.1. Construction of subgraphs connecting pairs of vertices.We will now construct for each un-ordered pair of cylindersC,D a subgraphLC,D of Ccyl that satisfies the conditions of Theorem 7.2with a constantM which will be derived along the way.

Let us first consider the caseC andD are parallel. IfC or D is non-degenerate thenι(C,D) = 0henced(C,D) = 1, which means thatC and D are connected by an edge inCcyl. We defineLC,D

to be this edge. If bothC andD are degenerate then it may happen thatι(C,D) > 0. Since (X, ω)is completely periodic, there is a non-degenerate cylinderE parallel toC and D. Sinceι(C,E) =ι(D,E) = 0, there are inCcyl two edges connectingE to C and toD. In this case, we defineLC,D to bethe union of these two edges.

Assume from now on thatC andD are not parallel. By applying an appropriate element of SL(2,R),we can assume thatC is horizontal,D is vertical, andC andD have the same circumference. For anyt ∈ R, set

at =

(et 00 e−t

)and (Xt, ωt) = at · (X, ω).

For any saddle connections in (X, ω), we will denote byℓt(s) its Euclidean length in (Xt, ωt). If E isa cylinder in (X, ω), thenct(E) andwt(E) are respectively its circumference and width in (Xt, ωt).

For anyR ∈ R>0, let L∗C,D(t,R) denote set of cylinders (possibly degenerate) of circumferencebounded above byR in (Xt, ωt). Note that this set is finite. Let us choose a constantL1 such that

(3) L1 > max 1K, 9,

and define

L∗C,D(L1) =⋃

t∈RL∗C,D(t, L1).

We regardL∗C,D(t,R) andL∗C,D(L1) as subsets ofC(0)cyl. Observe thatL∗C,D(t, L1) containsC when t

tends to−∞, and containsD whent tends to+∞, thereforeL∗C,D containsC andD.

26 DUC-MANH NGUYEN

For eacht ∈ R, consider now the setL∗C,D(t, 2L1). From Theorem 5.2,L∗C,D(t, 2L1) contains avertex corresponding to a cylinderC0,t of width bounded below byK. Set

(4) M1 := max2(2K1L1

K+ K2), 2

Then we haveLemma 7.3. As subset ofCcyl,L∗C,D(t, 2L1) has diameter bounded by M1.

Proof. Let E be a cylinder inL∗C,D(t, 2L1). If ι(E,C0,t) = 0, then we haved(C0,t,E) = 1. Otherwisewe haveKι(E,C0,t) ≤ ℓt(E) ≤ 2L1. Hence, from (1) we get

d(C0,t,E) ≤ 2K1L1

K+ K2,

and the lemma follows.

Moreover, we have

Lemma 7.4. Assume that the surface(X, ω) admits cylinder decompositions in both vertical andhorizontal directions. Then there exists a constant T> 0 such that

• if t > T thenL∗C,D(t, 2L1) only contains the vertical cylinders in(X, ω) andL∗C,D(t, 2L1) hasdiameter at most2,• if t < −T thenL∗C,D(t, 2L1) only contains the horizontal cylinders in(X, ω) andL∗C,D(t, 2L1)

has diameter at most2.

Proof. We only give the proof of the first assertion as the second one follows from the same argument.By assumption,X decomposes into the union of some non-degenerate vertical cylindersD1, . . . ,Dk.Let wt(Di) denote the width ofDi in (Xt, ωt). Let wt = minwt(Di), i = 1, . . . , k. A non-verticalcylinder must cross one ofDi, thus its circumference is bounded below bywt in (Xt, ωt). Since wehavewt = etw0, if t is large enough any non-vertical cylinder has circumference at least 2L1 in (Xt, ωt).HenceL∗C,D(t, 2L1) only contains the vertical cylinders. Since any vertical cylinder is of distance one

from D1 in Ccyl, L∗C,D(t, 2L1) has diameter at most two.

Lemma 7.5. If t− log(2)≤ t′ ≤ t+ log(2) thenL∗C,D(t′,R) ⊂ L∗C,D(t, 2R) for any R∈ R>0. In particularC0,t′ ∈ L∗C,D(t, 2L1).

Proof. Let s be a saddle connection or a regular geodesic in (Xt′ , ωt′). Let x + ıy be the period ofs in (Xt′ , ωt′). Note that (Xt, ωt) = at−t′ · (Xt′ , ωt′). Thus the period ofs in (Xt, ωt) is (et−t′ x, et′−ty).Therefore,

ℓt(s) =√

e2(t−t′ )x2 + e2(t′−t)y2 ≤ 2√

x2 + y2 = 2ℓt′ (s).

SetLC,D(2L1) :=

⋃

k∈ZL∗C,D(k log(2), 2L1) ⊂ C(0)

cyl.

It follows from Lemma 7.4 that ifn ∈ N is large enough then for anym > n, L∗C,D(m, L1) =L∗C,D(n, 2L1), andL∗C,D(−m, 2L1) = L∗C,D(−n, 2L1). Therefore, the setLC,D(2L1) is actually finite.

TRANSLATION SURFACES AND THE CURVE GRAPH IN GENUS TWO 27

For each unordered pair (x, y) of vertices inLC,D(2L1), let Γ(x, y) be a path of minimal length inCcyl

joining x to y. Set

LC,D(2L1) =⋃

x,y∈LC,D (2L1)

Γ(x, y).

As a direct consequence of Lemma 7.5, we get

Corollary 7.6.

a) If x ∈ L∗C,D(t, 2L1) and y∈ L∗C,D(t′, 2L1), thend(x, y) ≤ M1(2+|t−t′ |log(2)).

b) The setL∗C,D(L1) is contained inLC,D(2L1) andLC,D(2L1) is contained in the M1-neighborhoodofL∗C,D(L1).

c) For any pair of vertices(x, y) ∈ L∗C,D(L1)×L∗C,D(L1), there is a pathΓ(x, y) in LC,D(2L1) fromx to y of length equal tod(x, y).

7.2. Local property for LC,D. We will now show that the subgraphsLC,D(2L1) constructed abovesatisfy the first condition of Theorem 7.2.

Proposition 7.7. There exists a constant M2 such that if(X, ω) ∈ E then for any pair of cylinders C,Din (X, ω) such thatι(C,D) = 0, we havediamLC,D(2L1) ≤ M2.

To prove this proposition, we make use of an elementary result on slit tori (cf. Lemma B.3), andthe fact that ifC andD are not parallel, then there always exists a splitting ofX into two subsurfaces,each of which contains one ofC and D. Those auxiliary results are proved in the appendix. Themain technical difficulties arise when we have to deal with degenerate cylinders. We split the proof ofProposition 7.7 into two cases (X, ω) ∈ H(2) and (X, ω) ∈ H(1, 1).

Proof of Proposition 7.7, CaseH(2).

Proof. If C and D are parallel thenLC,D(2L1) has diameter bounded by 2 and we have nothing toprove. Suppose from now on thatC is horizontal,D is vertical,C andD have the same circumferenceequal toℓ, andLC,D(2L1) is the graph constructed above. Note that in this case (X, ω) is a Veechsurface, thus both horizontal and vertical directions are periodic.

Case 1:one ofC or D is non-degenerate. Assume thatC is non-degenerate. Letc be a core curve ofCandd a core curve ofD. Note thatc is a regular simple closed geodesic. By Lemma 3.6, the conditionι(C,D) = 0 implies thatc∩ d = ∅. Clearly,C cannot fillX. If C is not simple then the complementof C is a horizontal simple cylinderC′ whose boundary is contained in the boundary ofC. SinceDis disjoint fromC, it must be contained inC′. But this is impossible sinceC′ is horizontal andD isvertical. Therefore,C must be a simple cylinder.

The complement ofC is then a slit torus with the slit corresponding to the boundary of C. Remarkthat a core curve ofD must be disjoint from the interior of the slit, otherwise it would crossC entirely.Thus, we have in the slit torus an embedded square bounded by the boundary ofD and the slit (whichis actually the boundary ofC) (see Figure 7). By assumption, the length of the sides of this squareis ℓ. Since this square has area less than one, we must haveℓ < 1. ThereforeC ∈ L∗C,D(t, L1) for allt ≤ 0, andD ∈ L∗C,D(t, L1) for any t ≥ 0. Hence anyE ∈ LC,D(2L1) is of distance at mostM1 from C

or from D. Thus diamLC,D(2L1) ≤ 2M1 + 1.

28 DUC-MANH NGUYEN

D

C

Figure 7. Disjoint simple cylinders on surfaces inH(2)

Case 2:both ofC andD are degenerate. From Lemma 3.4, for anyǫ > 0 small enough, we can deform(X, ω) into another surface (X′, ω′) such that

• C corresponds to a simple horizontal cylinderC′ in X′ of width ǫ,• D corresponds to a vertical cylinder inX′.

Since ι(C′,D′) = ι(C,D) = 0, it follows from Lemma 3.6 thatD′ must be disjoint fromC′. Itfollows in particular thatD andD′ have the same circumferenceℓ. By constructionC′ has the samecircumference asC, and Area(X′, ω′) = Area(X, ω) + ǫℓ = 1 + ǫℓ. Applying the same arguments asabove to (X′, ω′), we see thatX′ contains an embedded square of sizeℓ disjoint fromC′. Thereforewe haveℓ2 < 1+ ǫℓ. Sinceǫ can be chosen arbitrarily, we derive thatℓ ≤ 1. We can then conclude bythe same arguments as the previous case.

Proof of Proposition 7.7, CaseH(1, 1).

Proof. Again, we only have to consider the caseC andD are not parallel. Thus we can assume thatCis horizontal andD is vertical. We first choose a positive real numberL >

√2 such that

(5) L1 ≥ 3 f (√

2L)

where f (x) =√

x2 + 1/x2 (see Lemma B.3).

Case 1:one ofC and D is a simple cylinder. By Lemma B.2, we need to consider two cases (seeFigure 8)

D

C

T

T′

E

D

C

s1

s2

s2P′

P′′

Figure 8. Disjoint cylinders on surfaces inH(1, 1): one ofC andD is simple.

TRANSLATION SURFACES AND THE CURVE GRAPH IN GENUS TWO 29

(i) There is a simple cylinderE disjoint from C ∪ D and the complement ofC ∪ D ∪ E is theunion of two triangles T,T′ (see Figure 8 left). Since we have Area(T)+ Area(T′) = ℓ2 <

Area(X, ω) = 1, it follows ℓ < 1. Hence we can use the same argument as in the case(X, ω) ∈ H(2) to conclude that diamLC,D(2L1) ≤ 2M1 + 1.

(ii) There is a pair of homologous saddle connectionss1, s2 that decomposeX into a connectedsum of two slit tori, (X′, ω′, s′) containingC and (X′′, ω′′, s′′) containingD (see Figure 8right).

By construction, the complement ofC in X′ is an embedded parallelogramP′ bounded bys1, s2 and the boundary ofC. Similarly, the complement ofD in X′′ is also an embeddedparallelogramP′′ bounded bys1, s2 and the boundary ofD. If ℓ ≤ 1 then we can concludeusing the argument above. Suppose that we haveℓ ≥ 1. Letω(si) = x + ıy. Since we haveArea(P′) = |y|ℓ, and Area(P′′) = |x|ℓ, it follows

max|x|, |y| ≤ 1/ℓ ≤ 1 and|si | =√

x2 + y2 ≤√

2/ℓ ≤√

2.

SetA1 = Area(X′, ω′), A2 = Area(X′′, ω′′), we haveA1 + A2 = 1. Without loss of generality,let us suppose thatA1 ≥ 1/2. For anyt ∈ R, the period ofsi in (Xt, ωt) is (etx, e−ty). Let(X′t , ω

′t , s′t) be the slit torus corresponding to (X′, ω′, s′) in (Xt, ωt). Recall that we have chosen

L >√

2 andL1 satisfies (5). Let us choose a positive real numberL′ ≥ 1 such that

L ≥√

L′2 + 1.

• For 0≤ t ≤ log(ℓL′), we haveet |x| ≤ L′ ande−t |y| ≤ |y| ≤ 1, thusℓt(s1) ≤√

L′2 + 1 ≤ L.Rescaling (X′t , ω

′t , s′t) by 1√

A1, we get a torus of area one with a slit of length bounded

by√

2L. Using Lemma B.3, we see that there exists in1√A1· X′t a cylinderE′t disjoint

from the slit of circumference bounded byL1. Note that inX′t , the circumference ofE′tis at most

√A1L1 ≤ L1. We haved(D,E′t ) = 1 andE′t ∈ L∗C,D(t, 2L1). Thus for any

E ∈ L∗C,D(t, 2L1) we haved(D,E) ≤ M1 + 1.

• For− log(ℓL′) ≤ t ≤ 0, we haveet |x| ≤ |x| ≤ 1 ande−t |y| ≤ L′, thusℓt(si) ≤√

L′2 + 1 ≤ L.The same argument as the previous case then shows thatd(D,E) ≤ M1 + 1, for anyE ∈ L∗C,D(t, 2L1).• For t ≥ log(ℓL′) we haveℓt(D) = e−tℓ ≤ 1/L′ ≤ 1 ≤ 2L1. ThusD ∈ L∗C,D(t, 2L1) which

implies thatd(D,E) ≤ M1 for anyE ∈ L∗C,D(t, 2L1).• For t ≤ − log(ℓL′) we haveℓt(C) ≤ 1/L′ ≤ 2L1, hence for anyE ∈ L∗C,D(t, 2L1),

d(C,E) ≤ M1, which implies thatd(D,E) ≤ M1 + 1.

We can then conclude that for anyt ∈ R, and anyE ∈ L∗C,D(t, 2L1), we haved(D,E) ≤ M1+ 1. Hence

diamLC,D ≤ 2(M1 + 1).

Case 2:one ofC,D is non-degenerate and not simple. Without loss of generality, we can assume thatC is neither simple nor degenerate. Lemma 3.6 implies thatD is disjoint fromC. SinceC is notsimple, the complement ofC is either (a) empty, (b) a horizontal simple cylinder, (c) the union of twosimple horizontal cylinders, or (d) another horizontal cylinder whose closure is a slit torus. Since thereexists a vertical cylinder disjoint fromC (namelyD), only (d) can occur. In this case, there are a pair

30 DUC-MANH NGUYEN

of horizontal homologous saddle connections1, s2 contained in the boundary ofC that decompose(X, ω) into the connected sum of two slit tori. Let (X′, ω′, s′) be the slit torus which is the closure ofC, and (X′′, ω′′, s′′) be the other one that containsD (see Figure 9).

D

C

s1

s2

s1

Figure 9. Disjoint cylinders on surfaces inH(1, 1): C is not simple nor degenerate.

Let x = |s1| = |s2|. Observe thatX′′ contains a rectangle bounded bys1, s2 and the saddle connec-tions borderingD. Therefore we havexℓ ≤ 1⇔ 0 ≤ x ≤ 1/ℓ. By the same arguments as the previouscase, we also get diamLC,D ≤ 2(M + 1).

Case 3:one ofC andD is degenerate. Let us assume thatC is degenerate. Using Lemma 3.4, wecan find a family (Xt, ωt), t ∈ [0, ǫ), of surfaces inH(1, 1) that are deformations of (X, ω), such thatC corresponds to a simple horizontal cylinderCt on Xt, for t > 0, which has the same circumference.Note that the width ofCt is t. Therefore Area(Xt, ωt) = Area(X, ω) + tℓ.

By construction,D corresponds to a cylinderDt on Xt which is disjoint fromCt (since we haveι(Ct,Dt) = ι(C,D) = 0). By Lemma B.2 we know that either (i) (Xt, ωt) contains two embeddedtriangles T,T′ disjoint fromCt andDt, or (ii) there is a splitting of (Xt, ωt) into two slit tori (X′t , ω

′t , s′t)

and (X′′t , ω′′t , s′′t ) such thatCt ⊂ X′t andDt ⊂ X′′t .

If (i) occurs, then we have Area(T)= Area(T′) = ℓ2/2 ≤ 1/2, which implies thatℓ ≤ 1. If (ii)occurs, then since the slits (s′ ands′′) are disjoint fromCt, they persist as we collapseCt to get back(X, ω). Thus, we have the same splitting on (X, ω). In conclusion, we can use the same arguments asin Case 1 to handle this case. The proof of Proposition 7.7 is now complete.

7.3. Slim triangle property for LC,D. We now prove that the subgraphsLC,D(2L1) satisfy the secondproperty of Theorem 7.2. The idea of the proof can found in [5,Lemma 4.4]. To alleviate the notations,in what follows we will writeLC,D instead ofLC,D(2L1).

Proposition 7.8. There exists a constant M3 such that for any triple of cylindersC,D,E in (X, ω),we haveLC,D is contained in the M3-neighborhood ofLC,E ∪ LE,D in Ccyl(X, ω, f ).

Proof. If C andD are parallel thenLC,D is contained in the 2-neighborhood ofLC,E ∪ LD,E. Fromnow on we assume thatC andD are not parallel.

By Corollary 7.6, we only need to show thatL∗C,D(L1) is contained in theM3-neighborhood of

L∗C,E(L1) ∪ L∗E,D(L1). Remark that to defineLC,D(2L1) andLC,D(2L1) one needs to specify an originfor the timet by the condition that the circumferences ofC andD are equal. On the other hand to

TRANSLATION SURFACES AND THE CURVE GRAPH IN GENUS TWO 31

defineL∗C,D(L1) this normalization is not required. IfE is parallel toC thenL∗C,D(L1) = L∗E,D(L1),and if E is parallel toD thenL∗C,D(L1) = L∗C,E(L1). In both of these cases we have nothing to prove.

Let us now assume thatE is neither parallel toC nor to D. We can then renormalize (usingSL(2,R)) such thatC is horizontal,D is vertical, andE has slope equal to one. Recall that for anyt ∈ R, (Xt, ωt) = at · (X, ω), C0,t is a cylinder of width bounded below byK in (Xt, ωt), and the constantL1 is chosen so thatL1 > 1/K (see (3)).

Claim: if t ≤ 0 thenC0,t is contained in theM1-neighborhood ofL∗C,E(L1).

Proof of the claim.Since (X, ω) is completely periodic, it decomposes into cylinders in both directionsof C andE. Let us denote byC = C1, . . . ,Cm the horizontal cylinders, and byE = E1, . . . ,En thecylinders in the direction ofE. As usual we denote byℓt(Ci) (resp. ℓt(E j)) the circumference ofCi

(resp. ofE j) in (Xt, ωt). Let ui(t) be the width ofCi, andv j(t) be the width ofE j in (Xt, ωt). Remarkthat

ℓt(Ci) = etℓ(Ci), ui(t) = e−tui , ℓt(E j) =√

cosh(2t)ℓ(E j), v j(t) =v j√

cosh(2t)Since (X, ω) has area one we have

(6) 1=∑

uiℓ(Ci) =∑

v jℓ(E j ).

Let x j (resp.yi) be the intersection number of a core curve ofC0,t and a core curve ofE j (resp. ofCi).Since the circumference ofC0,t is bounded by 1/K < L1, we have

(7)∑

yiui(t) = e−t∑

yiui ≤ ℓ(C0,t) ≤ L1⇒∑

yiui ≤ etL1.

Since the width ofC0,t is bounded below byK, we havex jK ≤ ℓt(E j) =√

cosh(2t)ℓ(E j ). Sincet ≤ 0, it follows

(8) x j ≤√

cosh(2t)K

ℓ(E j ) ≤e−t

Kℓ(E j).

Let (X′, ω′) := U · (X, ω), whereU =(

1 −10 1

). Let ℓ′(Ci) and u′i (resp. ℓ′(E j) and v′j) be the

circumference and the width ofCi (resp. ofE j) in (X′, ω′). Note thatCi is horizontal, andE j isvertical in (X′, ω′). Thus,ℓ′(Ci)) = ℓ(Ci), u′i = ui , andℓ′(E j) = ℓ(E j)/

√2, v′j =

√2v j .

For anys ∈ R, let (X′s, ω′s) := as · (X′, ω′). Let ℓ′s(Ci) andu′i (s) (resp. ℓ′s(E j) andv′j(s)) be the

circumference and the width ofCi (resp. ofE j) in (X′s, ω′s).

Let x+ ıy be the period of the core curves ofC0,t in (X′s, ω′s). From (8) we get

(9) |x| =∑

x jv′j(s) = es

∑x jv′j = es

√2e−t

K

∑ℓ(E j)v j =

√2es−t

K

From (7), we get

(10) |y| =∑

yiu′i (s) = e−s

∑yiui ≤ et−sL1.

Thus fors= t, the circumference ofC0,t in (X′s, ω′s) is at most

√3L1 < 2L1. Let C′0,s be a cylinder of

width bounded below byK in (X′s, ω′s). We haved(C′0,s,C0,t) ≤ M1 by Lemma 7.3, which means that

C0,t is contained in theM1-neighborhood ofL∗C,E(L1).

32 DUC-MANH NGUYEN

It follows immediately from the claim thatL∗C,D(t, L1) is contained in the 2M1-neighborhood ofL∗C,E(L1) if t ≤ 0. By similar arguments, one can also show thatL∗C,D(t, L1) is contained in the 2M1-neighborhood ofL∗E,D(L1) if t ≥ 0. Therefore, we can conclude thatL∗C,D(L1) = ∪t∈RL∗C,D(t, L1) is

contained in the 2M1-neighborhood ofL∗C,E(L1) ∪ L∗E,D(L1), which implies thatLC,D is contained inthe 3M1-neighborhood ofL∗C,E(L1) ∪ L∗E,D(L1).

7.4. Proof of Theorem 7.1.

Proof. From Proposition 7.7, and Proposition 7.8, we see thatCcyl(X, ω, f ) with the family of sub-graphsLC,D satisfies the two conditions of Theorem 7.2 withM = maxM2,M3. Therefore,Ccyl(X, ω, f )is Gromov hyperbolic.

8. Quotient by affine automorphisms

In this section we investigate the quotient ofCcyl(X, ω, f ) by the group Aff+(X, ω). Our main focusis the case where (X, ω) is a Veech surface, that is when SL(X, ω) is a lattice in SL(2,R). Throughoutthis section (X, ω) is a fixed translation surface inH(2) ⊔ H(1, 1), andCcyl is the cylinder graph of(X, ω) with some marking map. We denote byG the quotient graphCcylAff+(X, ω), and byV andE

the sets of vertices and edges ofG respectively. Notice that an edge may join a vertex to itself(we thenhave a loop), and there may be more than one edges with the sameendpoints. We use the notations|V | and|E | to designate the cardinalities ofE andV . We will show

Theorem 8.1. Let (X, ω) be a surface inH(2)⊔ H(1, 1). Then(X, ω) a Veech surface if and only if|V | is finite.

Theorem 8.1 does not mean, when (X, ω) is a Veech surface, that the quotient graphG is a finitegraph, as we have

Proposition 8.2. If (X, ω) is Veech surface inH(2) thenG is a finite graph, that is|V | and |E | areboth finite. There exist Veech surfaces inH(1, 1) such that|V | < ∞ but |E | = ∞.

8.1. Proof of Theorem 8.1. Recall that the SL(2,R)-orbit of a Veech surface (X, ω) projects to analgebraic curve inM2 isomorphic toX := HSL(X, ω), this curve is called aTeichmüller curve.The direction of any saddle connection onX is periodic, that isX is decomposed into finitely manycylinders in this direction. Moreover, there is a parabolicelement in SL(X, ω) that fixes this direction.Thus each cylinder inX corresponds to a cusp inX.

Let θ be a periodic direction forX. Let C1, . . . ,Ck be the cylinders ofX in the directionθ, andTi

be the Dehn twist about the core curves ofCi. Let γ be the generator of the parabolic subgroup ofSL(X, ω) that fixesθ. Then there exist some integersm1, . . . ,mk such thatγ is the differential of anelement of Aff+(X, ω) isotopic toTm1

1 · · · Tmkk .

8.1.1. Proof that(X, ω) is Veech implies thatV is finite.

Proof. If (X, ω) ∈ H(2), thenX has one or two cylinders in the directionθ. In the first case, we havethree more degenerate ones, and in the second case there is nodegenerate cylinder. Thus the totalnumber of cylinders (degenerate or not) in a periodic direction is at most 4. If (X, ω) ∈ H(1, 1), thenby similar arguments, we see thatX has at most 5 cylinders in the directionθ. We have seen thatθ corresponds to a cusp ofX. SinceX has finitely many cusps, it follows thatX has finitely manycylinders up to action of Aff+(X, ω). Therefore,V is finite.

TRANSLATION SURFACES AND THE CURVE GRAPH IN GENUS TWO 33

8.1.2. Proof thatV is finite implies(X, ω) is Veech.In what follows, by anembedded trianglein X, we mean the image of a triangleT in the plane by a

mapϕ : T → X which is locally isometric, injective in the interior ofT, and sending the vertices ofTto the singularities ofX. Note thatϕ maps a side ofT to a concatenation of some saddle connections.By a slight abuse of notation, we will also denote byT the image ofϕ in X. To show that (X, ω) is aVeech surface, we will use the following characterization of Veech surfaces by Smillie-Weiss [49].

Theorem 8.3(Smillie-Weiss). (X, ω) is a Veech surface if and only if there exists anǫ > 0 such thatthe area of any embedded triangleT in X is at leastǫ.

We now assume that|V | is finite. If v is a vertex ofCcyl, we denote by ¯v its equivalence class inV . Clearly, the group Aff+(X, ω) preserves the areas of the cylinders inX. Therefore, each elementof V has a well-defined area (a degenerate cylinder has zero area). SinceV is finite, we can writeV = v1, . . . , vn, wheren = |V |. Using GL+(2,R), we can normalize so that Area(X, ω) = 1. Letai = Area(vi), and define

A1 = a1, . . . , an,A2 = |ai − a j |, ai , a j,A3 = 1− (ai + a j), ai + a j < 1,A4 = 1− (ai + a j + ak), ai + a j + ak < 1.

Setǫ = minA1 ∪ A2 ∪ A3 ∪ A4. We will need the following lemma on slit tori.

Lemma 8.4. Let (X, ω, s) be a slit torus. By acylinder in X, we will mean a connected component ofX that is cut out by a pair of parallel simple closed geodesicspassing through the endpoints ofs.

Assume thats is not parallel to any simple closed geodesic ofX. Then there exists a sequence of

cylindersCkk∈N such thatCk is disjoint from the slits for all k ∈ N, and Area(Ck)k→+∞−→ Area(X).

Proof. Using GL+(2,R), we can normalize so that (X, ω) = (C/(Z ⊕ ıZ), dz). The slit s is thenrepresented by a segment [0, (1 + ıα)t], with t ∈ (0,∞) andα ∈ R \ Q. In this setting, each simpleclosed geodesicc of X corresponds to a vectorp+ ıq with p, q ∈ Z and gcd(p, q) = 1. Letc1 andc2

be the simple geodesics parallel toc which pass through the endpoints of ˆs. Note thatc1, c2 cut X intotwo cylinders. By [45, Lemma 4.1], we know that one of the two cylinders is disjoint from ˆs if andonly if

t|det(

p 1q α

)| = t|pα − q| < 1.

Note that the quantityt|pα − q| is precisely the area of the cylinder that contains ˆs. Sinceα is anirrational number, one can find a sequence(pk, qk)k∈N such that

gcd(pk, qk) = 1, t|αpk − qk| < 1, and limk→∞|αpk − qk| = 0.

For each (pk, qk) in this sequence, we have a cylinderCk in direction ofpk + ıqk disjoint from s suchthat

Area(Ck) = 1− t|αpk − qk|.In particular, we have limk→∞ Area(Ck) = 1, which proves the lemma.

As a consequence of this lemma, we get

34 DUC-MANH NGUYEN

Corollary 8.5. Let (s1, s2) be a pair of homologous saddle connections in X that are exchanged bythe hyperelliptic involutionτ. If one of the connected components cut out by(s1, s2) is a slit torus,then the direction of s1, s2 is periodic.

Proof. If (X, ω) ∈ H(2) thenX is decomposed by (s1, s2) into a simple cylinder and a slit torus, if(X, ω) ∈ H(1, 1) thenX is decomposed into two slit tori. Thus, it suffices to show thatsi is parallel toa closed geodesic in each slit torus. If this is not the case, then by Lemma 8.4, we can find in this slittorus a sequence of cylinders disjoint from the slit whose area converges to the area of the torus. Notethat such cylinders are also cylinders ofX. Thus their areas belong toA1. SinceA1 is finite, it cannotcontain a non-constant converging sequence. Therefore, wecan conclude that the direction of (s1, s2)is periodic.

Let T be an embedded triangle inX. We will show that Area(T) > ǫ/2. We first remark that itsuffices to consider the case where each side ofT is a saddle connection, since otherwise there isanother embedded triangle contained inT with this property. Letτ denote the hyperelliptic involutionof X, andT′ = τ(T). Let s1, s2, s3 be the sides ofT and s′i be the image ofsi by τ. The proof isnaturally splits into 2 cases depending on the stratum of (X, ω).

Case(X, ω) ∈ H(2): we need to consider the following two situations:

s1

s2 s3

s′1

TC1

Case 1

s1 s2

s3

s′1

s′1

s′2

s′2

C1

C2

T

Case 2

Figure 10. Embedded triangles in a surface inH(2).

• Case 1:none of the sides ofT is invariant byτ. From Lemma 2.4,si ands′i bound a simplecylinder denoted byCi. Let hi be length of the perpendicular segment from the oppositevertex ofsi in T to si . If int(T) ∩ int(C1) , ∅, then boths2 and s3 crossC1 entirely, whichimplies that the width ofC1 is is at mosth1 (see Figure 10 left). It follows that Area(T) ≥1/2Area(C1) > minA1/2. The same arguments apply in the cases int(T) intersects int(C2) orint(C3). If int(T) is disjoint from int(Ci), i = 1, 2, 3, then we have three disjoint cylinders inX (if int(Ci) ∩ int(C j) , ∅ thensi must crossC j entirely hence int(T) ∩ int(C j) , ∅). Since(X, ω) ∈ H(2), this situation cannot occur (see Theorem 2.6). Hence, we can conclude thatArea(T) ≥ ǫ/2 in this case.• Case 2:one of the sides ofT is invariant byτ. In this case, the union ofT and its image byτ is an embedded parallelogram (see Lemma 2.1). This means that there is a parallelogramP in the plane such thatT is one of the two triangles cut out by a diagonal ofP, and thereis a mapϕ : P → X locally isometric, injective in int(T), mapping the vertices ofP to thesingularity ofX. Remark that all the sides ofT cannot be invariant byτ because this would

TRANSLATION SURFACES AND THE CURVE GRAPH IN GENUS TWO 35

imply that X = ϕ(P) is a torus. If there are two sides ofT that are invariant byτ, thenϕ(P)is a simple cylinder inX, hence Area(T) ≥ minA1/2. If there is only one side invariantby τ, then the complement ofϕ(P) is the union of two disjoint simple cylindersC1,C2 (seeFigure 10 right), which implies Area(P) = 1 − (Area(C1) + Area(C2)). Therefore, we haveArea(T) > minA3/2 ≥ ǫ/2. This completes the of Theorem 8.1 for the case (X, ω) ∈ H(2).

Case(X, ω) ∈ H(1, 1): We consider the following situations:

s1

s2 s3

s′1

TC1 D1

P1

Case 1

C1

C2 C3

T

T′

Case 2

s1

s2

s3

s′2

s′3 DT

T′

Case 3

Figure 11. Embedded triangles in a surface inH(1, 1)

• Case 1:there existsi such thats′i intersects int(T). Note that we must haves′i , si . Let usassume thati = 1. Recall thats1 and s′1 either bound a simple cylinder, or decomposeXinto two tori. In the first case, the same argument as in the case (X, ω) ∈ H(2) shows thatArea(T) ≥ minA1/2. For the second case, observe that the intersection ofT with one ofthe slit tori consists of a domain bounded bys1 and some subsegments ofs2, s3 and s′1 (seeFigure 11). Let (X1, ω1, s1) denote this slit torus.

We can assume thats1 is horizontal. By Corollary 8.5 we know that the horizontal directionis periodic forX1, thusX1 is the closure of a horizontal cylinderC1. Remark thatX1 containsa transverse simple cylinderD1 disjoint from s1 ∪ s′1, whose core curves crossC1 once. Thecomplement ofD1 in X1 is an embedded parallelogramP1 bounded bys1, s′1 and the boundaryof D1. Clearly, we have Area(T) ≥ Area(P1)/2. By definition, we have

Area(P1) = Area(C1) − Area(D1) ≥ minA2.

Thus we have Area(T) ≥ ǫ/2.• Case 2:none ofs′i intersects int(T), ands′i , si , i = 1, 2, 3. It is not difficult to show that that

this case only happens whensi ands′i bound a simple cylinderCi disjoint from int(T)∪int(T′).Therefore,X is decomposed into the union of three cylindersC1,C2,C3, andT ∪ T′ (seeFigure 11). Thus in this case, we have

Area(T) =12

(1− (Area(C1) + Area(C2) + Area(C3))) ≥ minA4/2 ≥ ǫ/2.

• Case 3:none ofs′i intersects int(T) and one ofs1, s2, s3 is invariant byτ. Let us assume thats′1 = s1. It follows thatT ∪T′ is an embedded parallelogramP. If both (s2, s′2) and (s3, s′3) are

36 DUC-MANH NGUYEN

the boundaries of some simple cylindersC2 andC3 respectively, thenC2 andC3 are disjoint,andC2 ∪ C3 is disjoint fromP. By construction we must haveX = P ∪ C2 ∪ C3, which isimpossible since (X, ω) ∈ H(1, 1). Therefore, we can assume that (s2, s′2) decomposeX intotwo slit tori. Let X1 be the slit torus that containsP. By Corollary 8.5, we know that thedirection of (s2, s′2) is periodic, which means thatX1 is the closure of a cylinderC. Observethat the complement ofP in X1 must be a cylinderD bounded by (s3, s′3) (see Figure 11).Therefore

Area(T) =12

Area(P) =12

(Area(C) − Area(D)) ≥ 12

minA2 ≥ ǫ/2.

• Case 4:none ofs′i intersects int(T) and two ofs1, s2, s3 are invariant byτ. In this caseT ∪ T′

is a simple cylinder. Therefore Area(T) ≥ minA1/2 ≥ ǫ/2.

Since in all cases we have Area(T) ≥ ǫ/2, it follows from Theorem 8.3 that (X, ω) is a Veechsurface.

8.2. Proof of Proposition 8.2.

8.2.1. Case(X, ω) ∈ H(2).

Proof. We have shown thatV is finite, it remains to show thatE is also finite. Letv be a vertex ofCcyl, andC be the corresponding cylinder inX. We denote by ¯v the equivalence class ofv in G . UsingSL(2,R), we can suppose thatC is horizontal.

If C is a non-degenerate cylinder, then we have three cases: (a)C is the unique horizontal cylinder,(b) X has two horizontal cylinders andC is not simple, (c)C is a simple cylinder. In case (a), thereare 3 edges inCcyl that havev as an endpoint, those edges connectv to three degenerate cylinderscontained in the boundary ofC. In case (b), there is only one edge inCcyl havingv as an endpoint,this edge connectsC to the other horizontal simple cylinder. Thus in case (a) andcase (b), there areonly finitely many edges having ¯v as an endpoint.

Assume now that we are in case (c). LetD be the other horizontal cylinder ofX. Observe thatthe closure ofD is a slit torus (X′, ω′, s′) wheres′ corresponds to the boundary ofC. Let d be a corecurve ofD, ande be a simple closed geodesic inX′ disjoint from the slits’ and crossingd once. Weconsiderd, e as a basis ofH1(X′,Z). If C′ is a cylinder inX disjoint fromC, thenC′ must be entirelycontained inD. Thus the core curves ofC′ are determined by a unique element ofH1(X′,Z), and wecan writeC′ = md+ newith m, n ∈ Z.

By assumption, a core curvec′ of C′ cannot cross the slits′. The necessary and sufficient conditionfor this is that|ω′(c′) ∧ ω′(s′)| ≤ Area(X′) = Area(D) (see [45, Lem. 4.1]). But|ω′(c′) ∧ ω′(s′)| =|n||ω′(e) ∧ ω′(s′)|. Thus we can conclude that|n| is bounded by some constantn0.

We have seen that Aff+(X, ω) contains an elementφ = Tm11 Tm2

2 , whereT1 andT2 are the Dehntwists about the core curves ofC andD respectively. Observe thatφ fixes the vertices ofCcyl corre-sponding toC andD. The action ofφ on the curves contained inD is given by

φ(md+ ne) = (m±m2n)d + ne.

Thus up to action ofφkk∈Z, any cylinderC′ contained inD belongs to the equivalence class of acylinder C′′ also contained inD whose core curves are represented bymd+ nc with |n| ≤ |n0| and

TRANSLATION SURFACES AND THE CURVE GRAPH IN GENUS TWO 37

|m| ≤ |m2n| ≤ |m2||n0|. We can then conclude that there are finitely many edges inE which contains ¯vas an endpoint.

It remains to consider the caseC is degenerate. In this caseX has a unique non-degenerate cylinderin the horizontal direction, which containsC in its boundary. Remark that the complement ofCin X can be isometrically identified with a flat torus with an embedded geodesic segment removed.Therefore, the arguments above also hold in this case. Sincewe have proved that the set of vertices ofG is finite, it follows that the set of edges ofG is also finite.

8.2.2. Case(X, ω) ∈ H(1, 1).

Proof. Let (X, ω) the surface constructed from 6 squares as shown in Figure 12. This surface has 3horizontal cylinders denoted byC1,C2,C3, whereCi is the cylinder withi squares. It has two verticalcylinders denoted byD1 andD2, where the core curves ofD1 crossC1 andC3. Let v be the vertex ofCcyl corresponding toC1, andw be the vertex corresponding toC2. The fact thatG has finitely manyvertices follows from Theorem 8.1. We will show thatG has infinitely many edges.

C1

C3

C2

D1

D2

Figure 12. Example of square-tiled surface inH(1, 1)

Given a cylinderC on X, we denote byTC the Dehn twist about the core curves ofC. Observe thatf = T6

C1T3

C2T2

C3andg = TD1 T2

D2are two elements of Aff+(X, ω), their differentials are

(1 60 1

)and(

1 02 1

)respectively. Ifh is an element of Aff+(X, ω) that preserves the horizontal direction, thenh must

map a horizontal cylinder to a horizontal cylinder. SinceC1,C2,C3 have different circumferences,hmust preserve each of them, which implies thath = f k, k ∈ Z. We derive in particular that there is noaffine homeomorphism that mapsC2 to C1.

For anyn ∈ N, let En be the image ofC2 by gn. Remark thatEn = T2nD2

(C2), henceEn is contained in

the closureD2 of D2. In particular,En is disjoint fromC1. Thus, there is an edgeen in Ccyl connectingv to the vertexwn corresponding toEn. By definition, all the verticeswn belong to the equivalenceclassw of w in G . We will show that the edgesenn∈N are all distinct up to action of Aff+(X, ω), whichmeans that there are infinitely many edges inE between ¯v andw.

Assume that there is an affine automorphismh ∈ Aff+(X, ω) such thath(en1) = en2, for somen1, n2 ∈ N. If h(wni ) = v, then there is an element of Aff+(X, ω) that sendsw to v, or equivalentlyC2 to C1. But we have already seen that such an element does not exist,thus this case cannot occur.Therefore, we must haveh(v) = v andh(wn1) = wn2. Since any element of Aff+(X, ω) preservingC1 belongs to the subgroup generated byf , we derive thath also preservesC2 andC3. Observe thata core curve ofEni crossesC2 2ni times. Therefore, ifn1 , n2, thenh cannot exist. We can thenconclude that the projections of all the edgesen are distinct inG , which proves the proposition.

38 DUC-MANH NGUYEN

9. Quotient graphs and McMullen’s prototypes

By the works of McMullen [42, 39], we know that closed SL(2,R)-orbits inH(2) are indexed bythe discriminantD, that is a natural numberD ∈ N such thatD ≡ 0, 1 mod 4, together with the parityof the spin structure whenD ≡ 1 mod 8.

D = 5

C1

C2

√5−12

1

1

C2

C1

(0, 1,1,−1), λ =√

5−12

D = 8

√2

1

2

C1

C2

(0, 2,1,0)

√2− 1

1

1

C4

C3

(0, 1,1,−2)

C1

C2

C3

C4

D = 9

1

1

2

C1

C2

(0, 2,1,−1)

1

3

C3

C4

C3 is degenerate

C1

C2

C3

C4

Figure 13. Examples ofG for some small values ofD. For each two-cylinder decom-position, we provide the corresponding prototype (a, b, c, e). A loop at some vertexrepresents a Butterfly move that does not change the prototype.

Let (X, ω) be an eigenform inED ∩H(2) for some fixedD. Following McMullen [39], every two-cylinder decomposition ofX is encoded by a quadruple of integers (a, b, c, e) ∈ Z4 calledprototypesatisfying the following conditions

(PD)

b > 0, c > 0, gcd(b, c) > a ≥ 0,D = e2

+ 4bc, b > c+ e, gcd(a, b, c, e) = 1.

Setλ = (e+√

D)/2. Up to action of GL+(2,R), the decomposition ofX consists of two horizontalcylinders. The first one is simple and represented by a squareof sizeλ. The other one is non-simpleand represented by a parallelogram constructed from the vectors (b, 0) and (a, c). Note that we alwayshaveb > λ.

The quotient graphG turns out to be closely related to the set of McMullen’s prototypes. Namely,each prototype corresponds to a cluster of two vertices ofG which represent the cylinders in thecorresponding cylinder decomposition. LetC1,C2 be the cylinders in this decomposition, whereC1

is the simple one. Then the vertex corresponding toC2 is only adjacent to the one corresponding toC1 in G . This is because any other cylinder ofX must crossC2.

On the other hand, if there is an edge inG between two vertices representing two simple cylinderswhich are not parallel, then the two cylinder decompositions are related by a “Butterfly move” (see

TRANSLATION SURFACES AND THE CURVE GRAPH IN GENUS TWO 39

[39, Sect. 7] for the precise definitions). In other words,G can be viewed as a geometric objectrepresentingPD, each prototype is represented by two vertices connected byan edge, and all the otheredges ofG represent Butterfly moves.

There is nevertheless a slight difference between the two notions. The setPD only parametrizestwo-cylinder decompositions ofX, while in G we also have one-cylinder decompositions. If

√D < N,

then any cylinder inX is contained in a two-cylinder decomposition. Thus, the setof prototypesexhausts all the equivalence classes of cylinders inX (hence it provides the complete list of cuspsof the corresponding Teichmüller curve). But whenD is a square (e.g. D = 9), we need to takeinto account one-cylinder decompositions as well as degenerate cylinders. In Figure 13, we draw thequotient cylinder graphs of surfaces corresponding to somesmall values ofD.

Appendix A. Triangulations

In this section we construct triangulations of (X, ω) that are invariant by the hyperelliptic involution.The aim of these triangulations is to provide a “preferred” way to represent (X, ω) as a polygon inR2

when we have a horizontal simple cylinder onX. The results of this section are certainly not newand known to most people in the field (seee.g. [52]). We present them here only for the sake ofcompleteness.

In what follows, for any saddle connections, we will denote byh(s) the length of the horizontalcomponent ofs, that is h(s) = |Re(ω(s))|. If ∆ is a triangle bounded by the saddle connectionss1, s2, s3, we defineh(∆) = maxh(si), i = 1, 2, 3. Our main result in this section is the following

Proposition A.1. Let (X, ω) be a translation surface inH(2) ⊔ H(1, 1) having a simple horizontalcylinder C. Assume that every regular leaf of the vertical foliation of (X, ω) crosses C.

(i) If (X, ω) ∈ H(2), then (X, ω) can be obtained by identifying the pairs of opposite sides ofan octagonP = (P0 . . .P3Q0 . . .Q3) ⊂ R2 (see Figure 14), where the vertices are labelledclockwise, such that• −−−−−→PiPi+1 = −

−−−−−→QiQi+1, i = 0, 1, 2, and

−−−−→P3Q0 = −

−−−−→Q3P0,

• the diagonalsP0P3 and Q0Q3 are horizontal, the parallelogram(P0P3Q0Q3) is con-tained inP and projects to C⊂ X,• for i = 1, 2, the vertical line through Pi (resp. Qi) intersectsP0P3 (resp.Q0Q3), and the

vertical segment from Pi (resp. from Qi) to the intersection is contained inP.(ii) If (X, ω) ∈ H(1, 1), then(X, ω) can be obtained by identifying the pairs of opposite sides ofa

decagonP = (P0 . . .P4Q0 . . .Q4) (see Figure 14), where the vertices are labelled clockwise,such that• −−−−−→PiPi+1 = −

−−−−−→QiQi+1, i = 0, . . . , 3, and

−−−−→P4Q0 = −

−−−−→Q4P0,

• the diagonalsP0P4 and Q0Q4 are horizontal, the parallelogram(P0P4Q0Q4) is con-tained inP and projects to C⊂ X,• for i = 1, 2, 3, the vertical line through Pi (resp. Qi) intersectsP0P4 (resp. Q0Q4), and

the vertical segment from Pi (resp. from Qi) to the intersection is contained inP.

Proof. Cut off C from X, and identify the geodesic segments in the boundary of the resulting surface,we then obtain either a slit torus (if (X, ω) ∈ H(2)) or a surface inH(2) with a marked saddleconnection (if (X, ω) ∈ H(1, 1)). Let (X′, ω′) denote the new surface, ands′ the marked saddleconnection. If (X′, ω′) is a slit torus, then there is a unique involution ofX′ acting by−Id onH1(X′,Z)

40 DUC-MANH NGUYEN

s+

s−

P0

P1

P2

P3

Q0

Q1

Q2

Q3

(X, ω) ∈ H(2)

s+

s−

P0

P1

P2

P3

P4

Q0

Q1

Q2

Q3

Q4

(X, ω) ∈ H(1, 1)

Figure 14. Representations of surfaces inH(2) andH(1, 1) symmetric polygons.The simple horizontal cylinder is represented by the colored parallelogram.

and exchanges the endpoints ofs′. By a slight abuse of notation, we will call this involution thehyperelliptic involution ofX′. Thus, in both cases,s′ is invariant by the hyperelliptic involution.

By assumption all the regular vertical leaves ofX′ intersects′. Let ∆±i , i = 1, . . . , k be thetriangulation ofX′ provided by Lemma A.2 and Lemma A.3 (k = 2 if (X′, ω′) ∈ H(0, 0), k = 3 if(X′, ω′) ∈ H(2)). We can representC by a parallelogram inR2. The polygonP is obtained from thisparallelogram by gluing successively the triangles∆+1 , . . . ,∆

+

k , then∆−1 , . . . ,∆−k .

Lemma A.2. Let (X, ω, s) be a slit torus. Letτ be the elliptic involution of X that exchanges theendpoints P1,P2 of s. Assume that all the leaves of vertical foliation meet s.Then there exists aunique triangulation of X into4 triangles∆±1 ,∆

±2 with vertices inP1,P2, such that

• ∆+i and∆−i are exchanged byτ,• s is contained in both∆+1 and∆−1 ,• for i = 1, 2, the union∆+i ∪ ∆−i is a cylinder in X,• ∆+1 is adjacent to∆−1 and∆+2 , ∆−1 is adjacent to∆+1 and∆−2 ,• h(∆±1) = h(s), andh(∆±2 ) = h(c+), where c+ is the unique common side of∆+2 and∆+1 .

There are two possible configurations for this triangulation which are shown in Figure 15.

Proof. By Lemma 2.3, we know that there exists a pair of simple closedgeodesicsc+, c− passingthrough the endpoints ofs that cutX into 2 cylinders satisfyingh(c±) ≤ h(s). One of the cylinderscut out byc± containss, we denote it byC1, the other one is denoted byC2. Note that we must haveh(c±) > 0, otherwise there are vertical leaves that do not meets. It is easy to see that we get thedesired triangulation by adding some geodesic segments inC1 andC2 joining the endpoints ofs.

Lemma A.3. Let (X, ω) be a surface inH(2) and s be a saddle connection on X invariant by thehyperelliptic involutionτ. We assume that s is horizontal and all the leaves of the vertical foliationmeet s. Then we can triangulate X into 6 triangles∆±i , i = 1, 2, 3, whose sides are saddle connections,satisfying the following

• τ(∆+i ) = ∆−i , i = 1, 2, 3,

TRANSLATION SURFACES AND THE CURVE GRAPH IN GENUS TWO 41

s

∆+

1

∆−1

∆+

2

∆−2

P1 P2

P2

P1

P1

P2

s

∆+

1

∆−1

∆+

2

∆−2

P1 P2

P1

P2

P2

P1

Figure 15. Triangulation of a slit torus.

• ∆+1 and∆−1 contain s, andh(∆±1) = h(s),• ∆+2 has a unique common side with∆+1 which will be denoted by a+, andh(∆+2 ) = h(a+).• ∆+3 either has a unique common side b+ with ∆+1 and h(∆+3 ) = h(b+), or ∆+3 has a unique

common side c+ with∆+2 andh(∆+3 ) = h(c+).

This triangulation is unique. The configurations of the triangles∆±i , i = 1, 2, 3, are shown in Fig-ure 16.

∆+

1

∆−1

∆+

2

∆−2

∆+

3

∆−3

s∆+

1

∆−1

∆+

2

∆−2

∆+

3

∆−3

s∆+

1

∆−1

∆+

2

∆−2

∆+

3

∆−3

s

Figure 16. Triangulation of surfaces inH(2).

Proof. From Lemma 2.1, we see that there exist a parallelogramP ⊂ R2 and a locally isometric mapϕ : P→ X that maps a diagonal ofP to s. By construction,ϕ(P) is decomposed into two embeddedtriangles∆±1 , where∆+1 is the one aboves, both of which satisfyh(∆±1 ) = h(s) = |s|. Note also thatτ(∆+1 ) = ∆−1 .

Let us denote the non-horizontal sides of∆+1 by a+ andb+, and their images byτ by a− andb−

respectively. If both ofa+ andb+ are invariant byτ thenX = ϕ(P), which implies thatX is a torus,and we have a contradiction. Therefore, we only have two cases:

a) None ofa+, b+ is invariant byτ. In this cases, by Lemma 2.4 the complement ofϕ(P) isthe disjoint union of two cylinders bounded bya± andb± respectively. Note that none ofa+

andb+ is vertical, otherwise there would be vertical leaves that do not meets. We can thentriangulate the cylinders bounded bya± andb± in the same way as in Lemma A.2.

b) One ofa+, b+ is invariant byτ. We can assume thatb+ is invariant byτ. In this caseϕ(P)is actually a simple cylinder bounded bya±. The complement ofϕ(P) is then a slit torus

42 DUC-MANH NGUYEN

(X1, ω1, s1), wheres1 is the identification ofa±. From the assumption that all the verticalleaves meets, we derive thata± are not vertical. Thus we can follow the same argument as inLemma A.2 to get the desired triangulation.

Appendix B. Cylinders and decompositions

In this section, we give the proofs of some lemmas which are used in Section 7.

Lemma B.1. Let (X, ω) ∈ H(2) ⊔ H(1, 1) be a completely periodic surface in the sense of Calta.If C is a degenerate cylinder in X, then the direction of C is periodic, that is X is decomposed intocylinders in the direction of C.

Proof. If (X, ω) ∈ H(2) then (X, ω) is a Veech surface, thus the direction of any saddle connection isperiodic and we are done. Assume now that (X, ω) ∈ H(1, 1). InH(1, 1), we have a local action ofCwhich only changes the relative periods and leaves the absolute periods invariant. Orbits of this localaction are leaves of the kernel foliation. It is well known that the any eigenfom locus is invariant bythis local action.

Let us label the zeros ofω by x1, x2 and define the orientation of any path connectingx1 andx2 tobe fromx1 to x2. Using this local action ofC, we can collapse the two zeros ofω as follows, lets bea saddle connection invariant by the hyperelliptic involution satisfying the following condition:

(S) if there exists another saddle connections′ joining x1 andx2 such thatω(s′) = λω(s) withλ ∈ (0;+∞), then we haveλ > 1.

We can then reduce the length ofs to zero by moving in the kernel foliation leaf of (X, ω), theresulting surface is an eigenform inH(2) having the same absolute periods as (X, ω). The conditionon s implies thatx1 andx2 do not collide befores is reduced to a point. For a proof of this fact, werefer to [30, 31]. Remark that the new surface inH(2) is a Veech surface.

Without loss of generality, we can assume thatC is horizontal. By definition,C is the union of twosaddle connectionss1, s2 both invariant by the hyperelliptic involution, and up to a renumbering wehaveω(s1) ∈ R>0, ω(s2) ∈ R<0.

Assume that neither ofs1, s2 satisfies (S), then there exist two other saddle connectionss′1, s′2 such

thatω(s′i ) = λiω(si), with λi ∈ (0; 1). This implies that there are four horizontal saddle connectionson X. Since (X, ω) ∈ H(1, 1), there are at most 4 saddle connections in a fixed direction, and thismaximal number is realized if and only if the direction is periodic. Thus, in this case we can concludethatX is horizontally periodic.

Let us now assume that one ofs1, s2, say s1, satisfies the condition (S). We can then collapsex1, x2 along s1 to get a Veech surface (X0, ω0) ∈ H(2). Sinceω(s2) − ω(s1) is an absolute period, itstays unchanged along the collapsing procedure. Therefore, s2 persists inX0, and we haveω0(s2) =ω(s2) − ω(s1) ∈ R. In particular, (X0, ω0) has a horizontal saddle connection, and because (X0, ω0) isa Veech surface, it must be horizontally periodic. It follows that (X, ω) is also horizontally periodic.This completes the proof of the lemma.

Lemma B.2. Let (X, ω) ∈ H(1, 1). Let C be a horizontal (possibly degenerate) cylinder in X, and Dbe a vertical simple cylinder disjoint from C. Then either

TRANSLATION SURFACES AND THE CURVE GRAPH IN GENUS TWO 43

(a) there is another simple cylinder E disjoint from C∪D such that the complement of C∪D∪Eis the union of two embedded triangles, or

(b) there exist a pair of homologous saddle connections s1, s2 that decompose X into two two slittori (X′, ω′, s′) and(X′′, ω′′, s′′) such that C is contained in X′ and D is contained in X′′.

Proof. We first consider the caseC is not degenerate. In this case, the complement ofC in X is either(1) empty, (2) a horizontal simple cylinder, (3) the disjoint union of two horizontal simple cylinders,(4) a torus with a horizontal slit, or (5) a surface (X, ω) ∈ H(2) with a marked horizontal saddleconnections. Since we have a vertical simple cylinder disjoint fromC, only (4) and (5) can occur. Incase (4), we automatically have two slit tori, one of which isthe closure ofC, and the other one mustcontainD. Therefore we get case (b) of the statement of the lemma.

Let us now assume that we are in case (5). In this caseC must be a simple horizontal cylinder,and the saddle connections in X corresponds to the boundary ofC. Note thats is invariant by thehyperelliptic involutionτ of X. Let ϕ : P → X be the embedded parallelogram associated tos. Leta± andb± be the images byϕ of the sides ofP such thatτ(a+) = a− andτ(b+) = b−. Remark thatDmust be disjoint fromϕ(P) since any vertical geodesic intersectingϕ(int(P)) must intersect int(s), andhenceC, but we have assumed thatD is disjoint fromC.

If a+ = a− andb+ = b− then X must be a torus, and we have a contradiction. Therefore, we onlyhave two cases:

• a+ , a− and b+ , b−. In this case, the complement ofϕ(P) is the disjoint union of twosimple cylinders. SinceD is contained in this union,D must be one of the two. Let us denotethe other one byE. To obtain (X, ω) from (X, ω), we need to slit opens and glue backC.Consequently, we see that (X, ω) has three disjoint simple cylindersC,D,E. The complementof C∪D∪ E is the union of two embedded triangles, which are the images of the triangles inP cut out bys. Thus, we get case (a) of the statement of the lemma.• a+ = a− andb+ , b−. In this case,ϕ(P) is a simple cylinder bounded byb±. The comple-

ment ofϕ(P) is then a slit torus (X′′, ω′′, s′′) with the slit s′′ corresponding tob±. We canview (X′′, ω′′, s′′) as a subsurface ofX. Observe thatD must be contained in (X′′, ω′′) anddisjoint from the slits′′, since otherwise a core curve ofD must crossC. The complementof (X′′, ω′′, s′′) is another slit torus (X′, ω′, s′) which is obtained by slittingϕ(P) alongs andgluing backC. Therefore, we get case (b) of the statement of the lemma.

Assume nowC is degenerate. By Lemma 3.4, there exist deformations (Xt, ωt), t ∈ [0, ǫ), of (X, ω)such thatC corresponds to a simple horizontal cylinderCt in Xt, which has the same circumferenceasC and height equal tot. By construction,D corresponds to a simple vertical cylinderDt in Xt

which is disjoint fromCt. Observe thatCt andDt satisfy case (5) above. Therefore, by the precedingarguments, the conclusion of the lemma is true forCt and Dt. In either case, the correspondingdecomposition ofXt persists ast → 0, which implies that we have the same decomposition on (X, ω).

In what follows ifu = (u1, u2) andv = (v1, v2) are two vectors inR2, we denoteu∧ v := det( u1 v1

u2 v2

),

and|u|, |v| are the Euclidean norms ofu andv respectively.

Lemma B.3. Given a constant L> 0, let

(11) L1 := 3 max f (L), f (2δ)

44 DUC-MANH NGUYEN

where f(x) =√

x2 + 1/x2, andδ := (3/4)14 . Then for any slit torus(X, ω, s) with Area(X, ω) = 1, and

|s| < L, there exists in X a cylinder disjoint from s with area at least 1/2 and circumference boundedabove by L1.

Proof. Let Λ be the lattice inC such that (X, ω) can be identified with (C/Λ, dz). SinceΛ has co-volume one, there exists a vectorv ∈ Λ such that|v| ≤ δ. Setu = ω(s) ∈ C ≃ R2.

Let us first consider the case|u| ≤ 12δ . We then have

|u∧ v| ≤ |u||v| ≤ 1/2.

The vectorv corresponds to a simple closed geodesicc on X. The inequality above implies thatthere exist a pair of simple closed geodesics parallel toc cutting X into two cylinders, one of whichcontainssdenoted byC, the other one denoted byC′ consists of closed geodesics parallel toc that donot intersects (see [45, Lem. 4.1] or [39, Th. 7.2]). Note that the circumferences of bothC andC′ are|v| ≤ δ. Since Area(C) = |u∧ v| ≤ 1/2, we have Area(C′) ≥ 1/2. ThusC′ has the required properties.

We can now turn into the case12δ ≤ |s| ≤ L. By definition, we havef (|s|) ≤ L1/3. By multiplyingω by a complex number of module 1, which does not change the areaof X and the length ofs, wecan assume thats is horizontal. From Lemma 2.1, we know that there exists a local isometryϕ froma parallelogramP ⊂ R2 into X such that a horizontal diagonal ofP is mapped tos. SinceX is a torus,C := ϕ(P) is actually a cylinder inX. Letη be the distance from the highest point ofP to its horizontaldiagonal. By construction, we have Area(C) = Area(P) = η|s| ≤ Area(X, ω) = 1. Thusη ≤ 1/|s|.Remark that the boundary components ofC are the images byϕ of two opposite sides ofP. Hencethe circumference ofC is bounded by

√|s|2 + η2 ≤ f (|s|) ≤ L1/3.

Observe that the complement ofC is another cylinderC′ in X sharing the same boundary withC. If Area(C′) ≥ 1/2 then we are done. Let us consider the case Area(C′) < 1/2, which meansthat Area(C) > 1/2 > Area(C′). By cutting and pasting, we can also realizeC as a parallelogramQ = (P1P2P3P4) with two horizontal sidesP1P2 andP4P3 identified withs. Note that the distancebetweenP1P2 andP4P3 is η. We can then realizeC′ as a parallelogramQ′ = (P2P3P5P6) adjacent toQ whereP5 is contained in the horizontal stripe bounded by the lines supportingP1P2 andP4P3 (seeFigure 17). LetP′6 andP′5 be the intersections of the line supportingP5P6 and the lines supporting

P1P2 andP4P3 respectively.

P1 P2

P6

P′6

P5

P′5P3P4

Figure 17. Cylinder with bounded circumference and area≥ 1/2 in a slit torus.

Clearly we have Area(C′) = Area(Q′) = Area((P2P3P′5P′6)). Since Area(C′) < Area(C), wehave|P2P′6| < |P1P2|, and|P1P′6| < 2|P1P2| ≤ 2L. If P′6 ≡ P6, thenX has a horizontal cylinderC0

TRANSLATION SURFACES AND THE CURVE GRAPH IN GENUS TWO 45

with circumference equal|P1P′6| and area equal 1. Clearly the core curves ofC0 do not intersects,thereforeC0 has the required properties. IfP6 , P′6, then by construction,P1P5 andP4P5 project totwo simple closed geodesics inX, denoted byc1 andc2 respectively. These closed geodesics meetsonly at one of its endpoints. Letd1 andd2 be respectively the simple closed geodesics parallel toc1

andc2 passing through the other endpoint ofs. Observe thatc1 andd1 (resp.c2 andd2) cut X into 2cylinders, one of which containss will be denoted byC1 (resp.C2), the other one is denoted byC′1(resp.C′2). Now, remark that

Area(C1) = |−−−−→P1P5 ∧−−−−→P1P2|, and Area(C2) = |−−−−→P4P5 ∧

−−−−→P4P3|.

Since

|−−−−→P1P5 ∧−−−−→P1P2| + |

−−−−→P4P5 ∧

−−−−→P4P3| = |

−−−−→P1P2 ∧

−−−−→P1P4| = Area(C) ≤ 1,

we have either Area(C1) ≤ 1/2, or Area(C2) ≤ 1/2. Assume that Area(C1) ≤ 1/2, then Area(C′1) ≥1/2. Remark that

|c1| = |P1P5| ≤ |P1P′6| + |P′6P5| ≤ 2L1/3+ L1/3 = L1.

Thus we can conclude thatC′1 satisfies the statement of the lemma. In the case Area(C2) ≤ 1/2, thesame argument shows that the complementC′2 of C2 has the required properties. The proof of thelemma is now complete.

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E-mail address: duc-manh.nguyen@math.u-bordeaux1.fr